Engineering graphics

SVCE Mechanical Engg Dept ECE A
Tutorial Solutions- 2015 1
L ( Lecture Hours ) = 2
T ( Tutorial Hours ) = 0
P ( Practical Hours ) = 3
C ( Credits ) = 4
TOTAL: 75 PERIODS
A.Kumaraswamy
Assistant Professor
Department of Mechanical Engineering
Sri Venkateswara College of Engineering
GE6152 ENGINEERING GRAPHICS
A.KUMARASWAMY. AP, MECH
Cabin No : 4107
OBJECTIVES:
To develop in students, graphic skills for communication of concepts, ideas and design of
Engineering products.
To expose them to existing national standards related to technical drawings.
OUTCOMES:
On Completion of the course the student will be able to
Perform free hand sketching of basic geometrical constructions and multiple views of objects.
Do orthographic projection of lines and plane surfaces.
Draw projections and solids and development of surfaces.
Prepare isometric and perspective sections of simple solids.
Demonstrate computer aided drafting.

Publication Bureau of Indian Standards:
IS 10711 – 2001: Technical products Documentation – Size and lay out of
drawing sheets.
IS 9609 (Parts 0 & 1) – 2001: Technical products Documentation – Lettering.
IS 10714 (Part 20) – 2001 & SP 46 – 2003: Lines for technical drawings.
IS 11669 – 1986 & SP 46 – 2003: Dimensioning of Technical Drawings.
IS 15021 (Parts 1 to 4) – 2001: Technical drawings – Projection Methods.
Special points applicable to University Examinations on Engineering
Graphics:
There will be five questions, each of either or type covering all units of the
syllabus.
All questions will carry equal marks of 20 each making a total of 100.
The answer paper shall consist of drawing sheets of A3 size only. The students will
be permitted to use appropriate scale to fit solution within A3 size.
The examination will be conducted in appropriate sessions on the same day

CONCEPTS AND CONVENTIONS (Not for Examination) 1
Importance of graphics in engineering applications
Use of drafting instruments
BIS conventions and specifications
Size, layout and folding of drawing sheets
Lettering and dimensioning.
UNIT - I
PLANE CURVES AND FREE HAND SKETCHING 5+9
• Basic Geometrical constructions, Curves used in engineering practices:
Conics – Construction of ellipse, parabola and hyperbola by eccentricity
method – Construction of cycloid – construction of involutes of square and
circle – Drawing of tangents and normal to the above curves
• Scales: Construction of Diagonal and Vernier scales.
• Visualization concepts and Free Hand sketching: Visualization principles
–Representation of Three Dimensional objects – Layout of views- Free hand
sketching of multiple views from pictorial views of objects.

UNIT – II
PROJECTION OF POINTS, LINES AND PLANE SURFACES 5+9
• Orthographic projection- principles-Principal planes-First angle
projection
• Projection of points.
• Projection of straight lines (only First angle projections) inclined to
both the principal planes - Determination of true lengths and true
inclinations by rotating line method and traces
• Projection of planes (polygonal and circular surfaces) inclined to both
the principal planes by rotating object method.
UNIT – III
PROJECTION OF SOLIDS 5+9
• Projection of simple solids like prisms, pyramids, cylinder, cone and
truncated solids when the axis is inclined to one of the principal
planes by rotating object method and auxiliary plane method.

UNIT - IV
PROJECTION OF SECTIONED SOLIDS AND DEVELOPMENT OF
SURFACES 5+9
• Sectioning of above solids in simple vertical position when the cutting
plane is inclined to the one of the principal planes and perpendicular to the
other – obtaining true shape of section.
• Development of lateral surfaces of simple and sectioned solids – Prisms,
pyramids cylinders and cones. Development of lateral surfaces of solids
with cut-outs and holes
UNIT - V
ISOMETRIC AND PERSPECTIVE PROJECTIONS 6 + 9
• Principles of isometric projection – isometric scale –Isometric projections
of simple solids and truncated solids - Prisms, pyramids, cylinders, cones-
combination of two solid objects in simple vertical positions and
miscellaneous problems.
• Perspective projection of simple solids - Prisms, pyramids and cylinders by
visual ray method.

COMPUTER AIDED DRAFTING (Demonstration Only) 3
• Introduction to drafting packages and demonstration of their use.
TEXT BOOK:
• Bhatt N.D. and Panchal V.M., “Engineering Drawing”, Charotar Publishing House, 50th
Edition, 2010.

REFERENCE BOOKS:
• Gopalakrishna K.R., “Engineering Drawing” (Vol. I&II combined), Subhas Stores, Bangalore, 2007.
• Luzzader, Warren.J. and Duff,John M., “Fundamentals of Engineering Drawing with an introduction
to Interactive Computer Graphics for Design and Production”, Eastern Economy Edition, Prentice
Hall of India Pvt. Ltd, New Delhi, 2005.
• Shah M.B., and Rana B.C., “Engineering Drawing”, Pearson, 2nd Edition, 2009.
• Venugopal K. and Prabhu Raja V., “Engineering Graphics”, New Age International (P) Limited, 2008.
• Natrajan K.V., “A text book of Engineering Graphics”, Dhanalakshmi Publishers, Chennai, 2009.
• Basant Agarwal and Agarwal C.M., “Engineering Drawing”, Tata McGraw Hill Publishing Company
Limited, New Delhi, 2008.
Scheme of Evaluation
• Internal marks - 3 Continuous Assessment Tests
- Plates Submission ( weekly submissions )
- Model Examination
- 100 Marks
- University conversion into 20 Marks.
• University Examination (Theory)
- 100 Marks
- University conversion into 80 Marks.
There fore: Total = 20 + 80 = 100

Tutorial: II
Conic Sections and Space curves
Staff Incharge : A.Kumaraswamy
1
Construct a conic section, when the distance of
its focus from its directrix is equal to 40 mm
and its eccentricity as ¾. Name the curve and
draw a tangent and a normal to the curve at a
point P, 60 mm away from the directrix.

2
Draw the locus of a point which moves in such a
manner, so that its distance from a fixed point is
equal to its distance from a fixed straight line.
Take the distance between the fixed point and
the fixed line as 70 mm. Name the curve and draw
the tangent and normal at any point on it
70
3
The Vertex and the focus of a hyperbola are at
distances of 20 mm and 50 mm respectively
from a fixed straight line. Draw the hyperbola.
Draw a tangent and normal at any point on it.
50
20

4
Draw the locus of a point which moves in
such a manner, so that its distance from a
fixed straight line is 3/4 times the
distance from the fixed point. Take the
distance between the fixed point and the
fixed line as 70 mm. Name the curve and
draw the tangent and normal at any point
on it.
5 A circular wheel of diameter 60 mm rolls without slipping along a straight line. Draw the curve traced by a
point P on its rim for one revolution of the wheel. Draw the tangent and normal to the curve at a point,
located 35 mm above the line.

6
A rolling circle of 50 mm diameter
rolls on a fixed disc of 200 mm
diameter with external contact.
Draw the loci of path traced by a
point on the circumference of the
rolling circle for one complete
revolution. Draw the tangent and a
normal to the curve at a point 125
mm from the center of the base
circle.
7
A circle of 50 mm diameter rolls along the
inside of another circle of 180 mm
diameter. Draw the path described by a
point on the circumference of the rolling
circle for one complete revolution. Draw a
tangent and a normal at any point on the
curve .

8 Draw the involute of a pentagon of side 35 mm. Draw a tangent and a normal at any point on the curve.
9
Draw one turn of the
involute of a circle 60
mm in diameter. Draw a
tangent and a normal to
the curve at a point 80
mm from the center of
the circle.

Tutorial: III
Points and Staright Lines (Simple Cases)
Projections - Examples

Projections
Projections
Perspective Projection
• Centre of Projection (COP)
Parallel Projection
• Direction of Projection (DOP)
LOS
LOS
LOS – Line of Sight

ORTHOGRAPHIC PROJECTIONS - Principle.
• 'ORTHO' means right angle and orthographic means right angled drawing.
• When the projectors are perpendicular to the plane on which the projection
is obtained, it is known as orthographic projection.
• It is a parallel projection technique in which different views of an object are
projected on different reference planes observing perpendicular to respective
reference plane.
Reference planes or Principal planes
• Horizontal Plane (HP)
• Vertical Frontal Plane ( VP )
• Side Or Profile Plane ( PP)
Views in Orthographic projections
• Front View (FV) - FV is a view projected on VP.
• Top View (TV) - TV is a view projected on HP.
• Side View (SV) - SV is a view projected on PP.

Orthographic Rules: to draw on drawing sheet: 2 Dimensional
• Always Rotate the HP 90° through clockwise
• Always Rotate the PP 90° anti-clockwise for LSV
• Always Rotate the PP 90° clockwise for RSV
• OBJECT - Orientation and position will be described in the problem
• OBSERVER - Always standing in front of first quadrant vertical plane
• Terms ‘ABOVE’ & ‘BELOW’ with respective to H.P.
• Terms ‘INFRONT’ & ‘BEHIND’ with respective to V.P
X
Y
1ST Quad.2nd Quad.
3rd Quad. 4th Quad.
X Y
VP
HP
Observer
QUADRANT PATTERN

θ
Ø
A.V.P.
⊥ to Hp & ∠ to Vp
PLANES
PRINCIPAL PLANES
HP AND VP
AUXILIARY PLANES
Auxiliary Vertical Plane
(A.V.P.)
Profile Plane ( P.P.)Auxiliary Inclined Plane
(A.I.P.)
X
Y
X Y
VP
HP
PP
FV LSV
TV
PATTERN OF PLANES & VIEWS

Orthographic Projections
First Angle Projections Method
Object is in 1st Quadrant
( FV above X-Y, Tv below X-Y)
Third Angle Projections Method
object is in 3rd Quadrant.
( TV above X-Y, FV below X-Y)
FV
TV
X Y
X Y
G L
TV
FV
NOTE:-
HP term is used in 1st Angle method
&
For the same Ground term is used
in 3rd Angle method of projections
Symbolic Representation
NOTATIONS
IT’S FRONT VIEW a’ a’ b’
OBJECT POINT A LINE AB
IT’S TOP VIEW a a b
IT’S SIDE VIEW a” a” b”

Projection of points
HP
VP
a’
a
A
POINT “A” IN 1ST QUADRANT
OBSERVER
X HP Y
VP
m
n
n
a’
a
m

HP
VP
b’
b
B
POINT “B” IN 2nd QUADRANT
OBSERVER
X
HP
Y
VP
m
n
n
b’
b
m
b
HP
VP
c’
c
C
POINT “C” IN 3rd QUADRANT
OBSERVER
X Y
m
n
n
c’
c
m
c
VP
HP

HP
VP
d’
d
D
POINT “D” IN 4th QUADRANT
OBSERVER
X HP YVP
m
n
n
d’
d
m
d
HP
VP
a’
a
A
POINT “A” on HP and In front of VP
OBSERVER
X Ym
a’
a
m

HP
VP
a’
a
A
POINT “A” in VP above HP
OBSERVER
X Y
n
n
a’
a
1ST Quad.
2nd Quad.
3rd Quad. 4th Quad.
X Y
VP
HP
Observer
Point behind VP, above HP Point infront of VP, above HP
Point infront of VP, below HPPoint behind VP, below HP
Defining a point in different Quadrants

1. Draw the projections of the following points, keeping the distance between
the projectors as 30 mm on the common reference line.
A. Point A, 30 mm above HP and 25mm infront of VP.
B. Point B, in VP and 25 mm above HP
C. Point C, on HP and 30 mm infront of VP.
D. Point D, 25 mm above HP and 30 mm behind VP.
E. Point E, on HP and 40 mm behind VP.
F. Point F, 25 mm below HP and 40 mm behind VP.
G. Point G, on VP and 40 mm below HP
H. Point H, 30 mm below HP and 40 mm infront of VP.
I. Point I, is contained by both HP and VP.
30
a’
a
25
b
30
b’
25
POINT A POINT B POINT C
c
30
c’
30
d
25
d’
POINT D POINT E POINT F POINT G POINT H POINT I
e
e’
40
30 30 30 30 30 30 30
X Y
25
f’
40
f
g
40
g’
h’
40
h
30
i,i’

2
X Y
4040
b’
a,a’
b
a’b’ = 50 mm
A point A is on both HP and
VP. Another point B is 40 mm
above HP and 40 mm infront
of VP. Draw the projections
when the line joining their
plans is 50 mm. Also draw and
measure the line joining their
elevations
PROJECTIONS OF STRAIGHT LINES.
AIM:- TO DRAW IT’S PROJECTIONS – FV, TV and Traces

Required data to draw projections
• True length or apparent length
• Position of it’s ends with relevant to HP & VP
• It’s inclinations with respect to HP & VP
NOTATIONS
IT’S FRONT VIEW a’ b’
OBJECT LINE AB
TRUE LINE A B
IT’S SIDE VIEW a” b”
IT’S TOP VIEW a b
IT’S TRUE INCLINATION WITH HP θ
IT’S TRUE INCLINATION WITH VP φ
IT’S APPARENT INCLINATION WITH HP α
IT’S APPARENT INCLINATION WITH VP β

POSITIONS OF LINE
1. A LINE PERPENDICULAR TO HP AND PARALLEL TO VP
2. A LINE PERPENDICULAR TO VP AND PARALLEL TO HP
3. A LINE PARALLEL TO BOTH HP & VP.
4. A LINE INCLINED TO HP & PARALLEL TO VP.
5. A LINE INCLINED TO VP & PARALLEL TO HP.
6. A LINE INCLINED TO BOTH REFERENCE PLANES.
TRACES OF STRAIGHT LINES
The point of intersection of a line with a plane
Horizontal trace
The point of intersection of a line or line-produced with the HP is
termed the horizontal trace of the line usually represented by the letter HT.
Vertical trace
The point of intersection of a line or line-produced with the VP is known
as the vertical trace of the line denoted by the letters VT.

X
Y
a, (b)
a’
b’
B
A
TV
FV
X YH.P.
V.P.
a’
b’
A Line perpendicular to HP & Parallel to VP
Orthographic Projection
(Pictorial Presentation)
a, (b)
HT
HT
X
Y
a’, (b’)
a
B
A
TV
FV
X YH.P.
V.P.
a
b
A Line perpendicular to VP & Parallel to HP
a’, (b’)
b
VT
VT

X
Y
b’
a
B
TV
FV
X YH.P.
V.P.
a b
A Line Parallel to both HP and VP
b’
b
NO VT and HT
A
a’
a’
A Line inclined to HP and parallel to VP
X
Y
A
B
b’
θ
θ
a b
a’
b’
θ
a’
a
bh’
X YH.P.
V.P.h’
HT.
Apparent length
True length

A Line inclined to VP and parallel to HP
X
Y
a
b
a’ b’
X YH.P.
V.P.v
VT
b
b’
BA
φ
φ
a’
av
VT
φ
Apparent length
True length
3
A line AB, 75 mm long is parallel to
both HP and VP. The point A is 25 mm
above HP and point B is 40 mm infront
of VP. Draw its projections.

4
A line CD, 60 mm long is contained by HP and 20 mm infront of VP. Draw its projections.
5
A line PQ, 75 mrn long is contained by VP and 30 mrn above HP. Draw its projections.

6
A line EF, 70 mrn long is
perpendicular to HP and parallel to
VP. One end of the line E, is 15 mm
above HP and 30 mm infront of VP.
Draw its projections.
7
A line GH, has one of its end G on HP
and 15 mm infront of VP. The other
end H is 65 mm above HP. Draw its
projections, when the line is
perpendicular to HP and parallel to
VP.

8
A line IJ, 75 mm long is
perpendicular to VP and parallel to
HP. Draw its projections, when one
of its end I, is 15 mm above HP and
20 mm infront of VP.
9
A line MN, is perpendicular to VP
and parallel to HP. One end of the
line is 20 mm above HP, 25 mm
infront of VP and the other end is
75 mm infrorit of VP. Draw its
projections and determine its
length.

10
A line PQ, 65 mm long is inclined
at 45° to HP and parallel to VP. The
end P is 25 mm above HP and 20
mm infront of VP. Draw the
projections of the line PQ.
11
One end R of a line RS, is 20 mm
above HP and 25 mm infront of
VP. The other end S is 60 mm
above HP. Draw its projections
when the top view of the line
measures 60 mm. Also
determine the true length of the
line and the angle of inclination
of the line with HP.

12
A lineAB, 55 mm
long is parallel to HP
and inclined at 35° to
the VP. The end A is
20 mm above HP and
15 mm infront of VP.
Draw its projections
13
A line AB, 60 mm long
lies in HP and has its end
A in both HP and VP.
Draw its projections when
the line is inclined at 40°
to VP and parallel to HP.

Tutorial: IV
Staright Lines (Inclined to One Principle Plane)
UNIT – II
PROJECTION OF POINTS, LINES AND PLANE SURFACES 5+9
• Orthographic projection- principles - Principal planes - First angle
projection
• Projection of points
• Projection of straight lines (only First angle projections) inclined to
both the principal planes - Determination of true lengths and true
inclinations by rotating line method and traces.
• Projection of planes (polygonal and circular surfaces) inclined to both
the principal planes by rotating object method.

Drawing Practice:
• Maintain 100% cleanliness.
• Fix mini drafter properly and use it effectively – save more time.
• Show perfect distinction between thick and thin lines.
• Encircle Question Number with instrument (Ex: Pro-circle)
• Write question statement in both notebook and drawing sheet. Follow lettering practice in the
drawing sheet.
• Practice in Drawing notebook using basic instruments and reproduce in the drawing sheet (for
submission)
• Do not draw directly on the drawing sheet with out containing correct solution in the note
book.
• Practice all problems given in the tutorial book in the notebook.
• Draw only required number of problems on the drawing sheet prescribed by the instructor.
(interested students may submit all problems – imparts more practice)
• Draw proper dimensioning lines (maintain far distance from object line)
• Draw 1:3 type arrow head only.
• Do not erase thins lines anywhere.
POSITIONS OF LINE
1. A LINE PERPENDICULAR TO HP AND PARALLEL TO VP
2. A LINE PERPENDICULAR TO VP AND PARALLEL TO HP
3. A LINE PARALLEL TO BOTH HP & VP.
4. A LINE INCLINED TO HP & PARALLEL TO VP.
5. A LINE INCLINED TO VP & PARALLEL TO HP.
6. A LINE INCLINED TO BOTH REFERENCE PLANES.

Three methods to solve when a line inclined to both the principal
planes:
• Rotating Line Method
• Auxiliary Plane Method
• Rotating Trapezoidal Plane Method
NOTATIONS
IT’S FRONT VIEW a’ b’
OBJECT LINE AB
TRUE LINE A B
IT’S SIDE VIEW a” b”
IT’S TOP VIEW a b
IT’S TRUE INCLINATION WITH HP θ
IT’S TRUE INCLINATION WITH VP φ
IT’S APPARENT INCLINATION WITH HP α
IT’S APPARENT INCLINATION WITH VP β

Angle :
Possible to solve:
1. When θ + φ = 90° ( Projections will vertical straight lines on a single projector)
2. When θ + φ < 90° ( Projections will inclined straight lines on two projectors )
3. When θ + φ > 90° ( Impossible )
X
Y
a’
b’
a b
B
A
α
β
For Tv
T.V.
X Y
α
β
H.P.
V.P.
a
b
FV
TV
a’
b’
Pictorial Presentation Orthographic Projection

Rotating Line method:
Steps : For a true line AB
1. Assume the line is kept inclined to HP (θ) and parallel to VP. Draw a’b1’ and ab1 .
2. Assume the line is kept inclined to VP (φ) and parallel to HP. Draw ab2 and a’b2’ .
3. Draw the locus line parallel to reference line for the end B through b1’ and b2 .
4. Rotate the ab1 and a’b2’ to meet locus line. Mark the intersection point as b and b’.
5. Join the final projections ab and a’b’.
Cross check
1. a’ and a , b’ and b should lie on the same projector line….
2. The projector line joining a’ to a and b’ to b should be perpendicular to
reference line

Procedure to draw projections of a straight line inclined to both principal planes
• With given θ, draw true line with the inclination with
respect to HP. Mark the ends with a’ (above HP or on
HP) and b1’. Draw its topview parallel to VP and mark
as a and b1.
X YH.P.
V.P.
a’
b1’
a
θ
b1φ
b2
b2’
Locus of b
Locus of b’
b
b’
• With given φ, draw true line with the inclination
with respect to VP. Mark the ends with a (below
HP or on HP) and b2. Draw its frontview parallel
to HP and mark as a and b2’.
• Draw the Locus Lines for b’ and b
• Rotate the Top and Front views with a’ and a as
centers respectively till meeting the Locus lines.
α
β
• Draw Final front and top view projection
kl
mn
h’v
HTVT
1
A line AB, 75mm long is inclined
at 30° to HP and 45° to VP. The
end A is 25 mm above HP and 20
mm infront of VP. Draw the
projections of the line and also
locate its traces.

2
A line CD, 60 mm long, has one
of its ends C on VP and 25 mm
above HP. Draw the projections of
the line CD if it makes 30° to HP
and 40° to VP. Also locate the
vertical and horizontal traces.
3
One end A of a line AB, 75 mm
long is in both HP and VP. The
line is inclined at 40° to the HP
and 35° to the VP. Draw its
projections.

4
A straight line 85 mm long has
one end 15mm infront of VP and
10 mm above HP, while the other
end is 50 mm infront of VP and
45 mm above HP. Draw the plan
and elevation of the line.
Determine the inclinations of the
line to Hpa nd VP. Locate the
traces of the line.
5
A straight line 70 mm long has
one end 15 mm infront of VP and
50 mm above HP, while the other
end is 35 mm infront of VP and
20 mm above HP. Draw the plan
and elevation of the line and
determine its traces.

6
A line AB 70 mm long has its end
B, 25 mm above HP and 30 mm
infront of VP. The end A is 55 mm
away from the reference planes.
Draw the projections and find its
inclinations with VP and HP.
7
The end P of a line PQ is 50 mm
away from both the reference
planes and the other end Q is
20 mm above HP and 25 mm
infront of VP. Draw the
projections of the line PQ and
find its true inclinations with
HP and VP if the length of the
line is 60 mm.

8
A line AB of 75 mm long has
one of its ends 60 mm infront of
VPand 20 mm above HP. The
other end is 20 mm infront of
VP and is above HP. The top
view of the line is 55 mm long.
Draw its front view. Also locate
its traces.
9
The end P of a line PQ, 70 mm
long is 15 mm above HP and 20
mm infront of the VP. The end Q is
40 mm above HP and the top view
of the line is inclined at 450 to the
VP. Draw the projections of the
line and find its true inclinations
with VP and HP.

10
End A of the line AB of 75 mm
long is 50 mm infront of VP and
15 mm above HP. End B is 15 mm
infront of VP and above HP. The
top view of the line is measured
to be 50 mm long. Determine the
length of its elevation and the
true inclination of the line.
11
The top view of a line PQ makes
an angle of 300 with the horizontal
and has a length of 100 mm. The
end Q is in the HP and P is in the
VP and 65 mm above the HP.
Draw the projections of the line
and find its true length and true
inclinations with the reference
planes. Also show its traces.

12
A line measuring 80 mm long has
one of its ends 60 mm above HP
and 20 mm infront of VP. The
other end is 15 mm above HP and
infront of VP. The frontview of
the line is 60 mm long. Draw its
topview. Also locate its traces.
13
The front view of line AB, 90 mm
long measures 65 mm and is
inclined at 450 to HP. Its one end A
is 15 mm above HP and on VP.
Draw the projections of the line
and find its inclinations with HP
and VP.

14
A line PQ 60 mm long has its end
P, 15 mm above HP and 20 mm
infront of VP. Its top and front
view measures 50 mm and 40 mm
respectively. Draw its projections
and determine the true
inclinations with HP and VP
15
A line AB 60 mm long, has its end
A 30 mm above HP and 25 mm
infront of VP. The top view and
front view has a length of 40 and
55 mm respectively. Draw its
projections.

16
A line EF, 85 mm long has its end E,
25 mm above HP and 20 mm infront
of VP. The plan and elevation of the
line measures 55 mm and 70 mm
respectively. Draw the projections of
the line and find its true inclinations
with HP and VP.
17
The plan and elevation of a line
AB are inclined at 35° and 50° to
the XY line respectively. One end
of the line is touching both HP
and VP. The other end is 50mm
above HP. Find its true length and
true angle of inclinations with HP
and VP.

18
A line PQ has one of its
extremities 15 mm above HP and
10 mm infront of VP. The end Q is
55 mm above HP and line is
inclined at 300 to HP. The distance
between the end projectors of the
line when measured parallel to the
line of intersection of HP and VP
is 50mm. Draw the projections of
the line and find its true length
and true inclination with VP.
19
End A of the line AB is 15 mm
above HP and 20 mm infront of
VP. The other end is 50 mm
above HPand 65 mm infront of
VP.The distance between the
end projectors is 50 mm. Draw
the projections and find the
true inclinations and true
length of the line.

20
The distance between the end
projectors passing through the end
points of a line AB is 60 mm.The end A
is 15 mm above HP and 10 mm infront of
VP. The end B is 35mm infront of VP.
The line AB appears 70 mm long in the
front view. Complete the projections by
trapezoidal methodand find the true
length of the line and its inclinations
with HP and VP. Also locate its traces.
21
A line RS, 65 mm long has
its end R, 15 mm above the
HP and 15 mm infront of
VP. The line is inclined at
550 to the HP and 350 to the
VP. Draw its projections.

22
Draw the projections of a
straight line AB, 100 mm
long, inclined at 450 to HP
and 300 to VP. The end A is in
HP and the end B is in VP.
23
A line 100 mm long has its lower end in
HP and the upper end in VP. Its plan
and elevation measure 80 mm and 70
mm respectively. Draw the projections
of the line and find its True
inclinations with HP and VP.

24
The mid point of line AB, 80 mm long is 80 mm
above HP and 50 mm infront of VP. The line is
inclined at 300 to HP and 450 to VP. Draw its
projections.
25
A line AB of length 80 mm is inclined
at 45° to HP and 30° to VP. The point C
is on the line which situated 20 mm
from the end A and is 30 mm above HP
and 40 mm infront of VP. Draw the
projections of the line.

26
The distance between the projectors
through the VT and the HT of a line
PQ is 70 mm and that between the
projectors through the ends is 40 mm.
The VT is 45 mm above the HP and
the HT is 30 mm in front of the VP. P
is 15 mm above the HP. Draw the front
and top views of the line. Also find
the true length and the inclinations of
the line with the HP and the VP.
27
The distance between the
projectors of a line CD is 70 m. Its
ends Cand D are in front of the
VP at 60 mm and 30 mm
respectively from the VP. The HT
of the line is 15 mm in front of the
VP. The line is inclined at 35° to
the HP. Draw the projections of
the line and find its true
inclination with the VP. Also
locate its VT.

28
A line PQ has its ends 10 mm and 45
mm above the HP and the length of
its front view is 70 mm. The line is
inclined at 25° to the HP. The HT of
the line is 15 mm in front of the VP.
Draw the projections of the line and
find its true length and true
inclination with the VP. Also, show
its VT.
9
The end P of a line PQ, 70 mm
long is 15 mm above HP and 20
mm infront of the VP. The end Q
is 40 mm above HP and the top
view of the line is inclined at 450
to the VP. Draw the projections of
the line and find its true
inclinations with VP and HP.
X YH.P.
V.P.
p’
p
2015
40
45°
q1
q
Locus of q
Locus of q’
φ
q2
q2’
q’ q1’
θ
Θ = 20°
Φ = 40°

Tutorial: V
Projections of Plane Surfaces
In Syllabus:
Projection of planes (polygonal and circular surfaces) inclined to
both the principal planes by rotating object method.
PROJECTIONS OF PLANE SURFACES

Projections of Planes
• A plane is a two dimensional object having length and breadth only.
• Thickness is negligible.
Types of Planes
1. Perpendicular plane which have their surface perpendicular to
any one of the reference planes and parallel or inclined to the
other reference plane.
2. Oblique plane which have their surface inclined to both the
reference planes.
Positions of a plane
Perpendicular planes
Surface of the plane kept perpendicular to HP and parallel to VP
Surface of the plane kept perpendicular to VP and parallel to HP
Surface of the plane kept perpendicular to both HP and VP
Surface of the plane kept inclined to HP and perpendicular to VP
Surface of the plane kept inclined to VP and perpendicular to HP
Oblique plane
Surface of the plane kept inclined to both HP & VP

Traces of planes
A plane, extended if necessary, will meet the reference planes in lines unless it
is parallel to any one of them. These lines are called traces of planes.
The line in which the plane meets the H.P is called the horizontal trace or H.T
of the plane.
The line in which the plane meets the V.P is called the vertical trace or V.T of
the plane.
Plane surfaces
Square
Triangle
Pentagon Hexagon
Rhombus
Rectangle
Rhomboid
Trapezoid Trapezium
ETC……
Heptogon Octagon
Parallelograms

Simple Position
Points to remember :
• When a plane surface is parallel to HP, top view shows its true shape &
size. Therefore, draw the top view first. Project the front view from it.
• When a plane surface is parallel to VP, front view shows its true shape
& size. Therefore, draw the front view first. Project the top view from
it.
A
B
C
D
(b), a
(c), d
b'
a'
d'
c'
X HP
VP
Y
a'
b' c'
d'
(b), a (c), d
NO VT
HT
Surface of the plane kept perpendicular to HP and parallel to VP

A
B
C
D
b’,(a’)
c’,(d’)
b
a
d
c
X HP
VP
Y
a d
b c
b’,(a’) c’,(d’)
NO HT
VT
Surface of the plane kept perpendicular to VP and parallel to HP
A
B
C
D
(b), a
(c), d
(b’)c’
(a‘)d’
X HP
VP
Y
(b), a
(c), d
(b’)c’
(a‘)d’
VT
HTVT
Surface of the plane kept perpendicular to both HP and VP

a’
b’
d’
c’
A
B
C
Φ
Φ
b(c)
a(d)
D
X HP
VP
Y
a(d)
b(c)
a’ b’
d’ c’
VT
Φ
VT
Surface of the plane kept inclined to VP and perpendicular to HP
a’(d’) b’(c’)
a
b
c
d
A
B
CD
ϴ
ϴ X HP
VP
Y
a b
cd
ϴ
a’(d’)
b’(c’)
HT
Surface of the plane kept inclined to HP and perpendicular to VP

Plane surfaces inclined to both HP and VP
Change of position method :
• Orthographic projections of a plane surface inclined to one reference plane are
drawn by changing the position of the plane surface from simple position to the
required position in two stages known as the change of position method.
Three steps are given below:
1. Simple position
2. Second position
3. Final position
HP
VP
VPVP
a’ d’
c’b’
HP
a
b c
d
a1’
d1’ c1’
b1’
a1
b1 c1
d1
SURFACE PARALLEL TO HP
PICTORIAL PRESENTATION
SURFACE INCLINED TO HP
ONE SMALL SIDE INCLINED TO VP
ORTHOGRAPHIC
TV-True Shape
FV- Line Parallel to xy
ORTHOGRAPHIC
FV- Inclined to XY
TV- Reduced Shape
ORTHOGRAPHIC
FV- Apparent Shape
TV-Previous Shape
STEP I STEP II STEP III

X YHP
VP
30°
a
b
c
d
20
□40
a’ b’ c’d’
A Square lamina of side 40 mm has its
surface parallel to and 20 mm above
HP. Draw its projections when one of
its sides is inclined at 300 to the VP.
1
A.KUMARASWAMY. AP, MECHTutorial: V
A Hexagonal plate of 30 mm side has
its surface parallel to VP and 20 mm
infront of it. Draw the projections
when one of its sides is perpendicular
to HP.
2

X YHP
VP
25°
a’
c’
b’
e’
d’
30
a e cd(b)
A Pentagonal plane of side 30 mm is
parallel to VP and perpendicular to HP.
Draw its projections when one of its corners
is on HP and the side containing the resting
corner is inclined at 250 to HP.
3
X YHP
VP
a a1
b
c
b1
c1
d d1
e e1
f f1
a’(f’) b’(e’) c’(d’) a1’(f1’)
c1’(d1’)
b1’(e1’)
45°
20
30
A Hexagonal plate with 30 mm
side has its surface perpendicular
to VP and inclined at 450 to HP.
Draw its projections when one of
its sides is on HP and the corner
nearer to the VP is 20 mm
infront of it.
4

X YHP
VP
Φ60
40
35
a h(b) g(c) f(d) e
a’
c’
b’
d’
e’
f’
g’
h’
a1
h1(b1)
g1(c1)
f1(d1)
e1
40°
a1’
c1’
b1’ d1’
e1’
f1’
g1’
h’
A Circular plane of 60 mm
diameter has its plane vertical
and inclined at 400 to VP. Its
centre is 40 mm above HP and
35 mm infront of VP. Draw its
projections.
5
A 60° Set square has its shortest edge of length 40 mm kept
perpendicular to the VP so that the projection of the set square on
the HP is an isosceles triangle of side 40 mm. Draw the projections
of the set square and find the inclination of the set square with the
HP.
6

An equilateral triangular lamina of side 50 mm has one of its edge on HP inclined at 40° to
the VP. Draw its projection when the surface is inclined at 30°to HP.
7
8
A square plate of side 35 mm rests on the ground on one of its sides inclined at 40° to
the VP. Its surface makes 50°with the ground. Draw its projections.

9 A Rectangular plate 70 X 40 mm has one of its shorter edges in the HP inclined at 40° to the
VP. Draw its front view if its top view is a square of side 40mm
10 A Pentagonal plate of side 25 mm has one of its sides on VP inclined at 45° to the HP. Draw
its projections when the plane surface is inclined at 30°to VP.

11 A Hexagonal plate of side 20 mm rests on the HP on one of its sides inclined at 450 to the
VP. The surface of the plate makes an angle of 300 with the HP. Draw its front and top views.
12 An equilateral triangular plate PQR has 50 mm sides. It rests on the ground on one of its
edges making an angle of 45° with the VP. The other edges have length of 40 mm each in the
top view. Draw the projections of the plate and find its inclinations with the HP.

13
An equilateral triangle ABC of side 60mm is resting on the HP on one of its edge inclined at 45° to the VP.
The plane is tilted to HP in such a way that the difference in the height between the resting edge and the
corner opposite to the resting edge is 30 mm. Draw the projections of the plane.
X YHP
VP
50
70
40
40°
30°
q
rp
q’(p’) r’
q1’(p1’)
r1’
q1
r1
p1
q2
r2
p2
q2’
p2’
r2’
14
A Triangular plate PQR has sides PQ = 50 mm, QR = 70 mm and RP = 40 mm. The side PQ rests on HP and is
inclined at 300 to the VP. The surface of the plate is inclined at 400 to the HP. Draw the projections of the
triangular plate.

15
An isosceles triangular plate of side 150 mm and altitude 225 mm appears like an equilateral triangle of side 150 mm in the top
view. Draw the projections of the plate if its 150 mm long edge is on HP inclined at 45° to the VP. Also find the inclination of the
plate with the HP.
16
A hexagonal lamina of side 35mm rests on one of its edges on the HP. This edge is parallel to the VP. The surface of the lamina is
inclined at 60° to the HP. Draw the projections.

17
A Square lamina PQRS of side 40 mm rests on
the ground on its corner P in such a way that the
diagonal PR is inclined at 45° to the HP and
apparently inclined at 30° to the VP. Draw its
projections.
18
A Square lamina PQRS of side 40 mm rests on
the ground on its corner P in such a way that the
diagonal PR is inclined at 45° to the HP and
apparently inclined at 30° to the VP. Draw its
projections.

19
A Pentagon of side 30 mm rests on the ground on one of its corners with the sides containing the corner
being equally inclined to the ground. The side opposite to the corner on which it rests is inclined at 30o
to VP and is parallel to HP. The surface of the pentagon makes 50o with the ground. Draw the top and
front views of the pentagon.
20
A circular lamina of diameter 80 mm is resting on the HP on one of its circumferential points. Draw its projections when its top
view is an ellipse of minor axis, 40mm and the major axis is parallel to VP

p2
45°
A Circular plate of diameter 80 mm has the end P of the diameter PQ in the HP and the plate is
inclined at 45o to HP. Draw the projections when
1. The diameter PQ appears to be inclined at 45o to VP in TV.
2. The diameter PQ makes 45o with VP.
X YHP
VP
Φ80
1
2 3
s
q
4
5
6
r
7
8
p
p’ s’(r’) q‘
11
21
31
s1
q1
41
51
r1
71
81
p1
61
32
p1’
s1’(r1’)
q1’
32’
42’
q2’
52’
62’
r2’
72’
82’p2’
12’
22’
s2’
42
q2
52
62
r2
72
82
12
22
s2
45°
q
p2q - Actual diameter
P2q2 – Apparent Diameter
21
Solution for part
2 only.
Part 1: Step 2 can
be extended in
usual procedure
to obtain step 3
22
A Rectangular plate 70 X 40 mm has one of its shorter edges in the VP inclined at 400 to the HP. Draw its top view if its front
view is a square of side 40 mm.

X YHP
VP
70 □40
40
a b
cd
a’(d’) b’(c’)
d1
a1
b1
c1
a1’(d1’)
b1’(c1’)
40°
a2
b2
c2
d2
d2’ a2’
b2’c2’
EXAMPLE
A Rectangular plate 70 X 40 mm has one of its shorter
edges in the HP inclined at 400 to the VP. Draw its
front view if its top view is a square of side 40 mm
X YHP
VP
25
a’
b’
c’
d’
e’
e(a) d(b) c e1(a1)
d1(b1)
c1
a1’
b1’
c1’
d1’
e1’
30°
45°
e2
b2
c2
d2
a2
a2’
b2’c2’
d2’
e2’
23
A Pentagonal plate of side 25 mm has one of its side on VP inclined at 450 to the HP. Draw its
projections when the plane surface is inclined at 300 to VP.

24
A Hexagonal plate of side 20 mm rests on the VP on one of its sides inclined at 45° to the HP.
The surface of the plate makes an angle of 30° with the VP. Draw its front and top views.
25
A Square lamina PQRS of side 40 mm rests on the VP on its corner P in such a way that the diagonal PR is
inclined at 45° to the VP and apparently inclined at 30° to the HP. Draw its projections.

26 A hexagonal plate of side 30 mm rests on the VP on one of its corners with the sides containing the corner
being equally inclined to the VP. A diagonal passing through the resting corner is inclined at 40°to the HP.
The surface of the plate makes 50° with the VP. Draw the top and front views of the hexagonal plate.
c2
X YHP
VP
35
35°
a
b c
d
ef
a’ b’(f’) c’(e’) d’
a1
b1
c1
d1
e1f1
a1’
b1’(f1’)
c1’(e1’)
d1’
55°
a2
b2
d2
e2
f2
a2’
b2’
c2’
d2’
e2’
f2’
27
*** ϴ + Φ = 90°
A regular Hexagonal plate of 35 mm side has one corner touching VP and another opposite corner
touching HP. The plate is inclined at 550 to HP and 350 to VP. Draw the projections of the plate.

28
A hexagonal lamina of side
30 mm is resting on the HP
in such a way that one of its
corners is touching both HP
and VP. Draw its projections
when its surface makes 30°
with the HP and 60°with the
VP.
29 A Circular lamina of diameter 60 mm has one end of its diameter in HP and other end of the
diameter in VP. The lamina is inclined at 30° to HP and 60° to VP. Draw the projections of
the lamina.

A Hexagonal Plate of side 30 mm rests on the ground
on one of its corners with the sides containing the
corner being equally inclined to the ground. A
diagonal passing through the resting corner is
inclined at 400 to the VP. The surface of the plate
makes 50o with the ground. Draw the top and front
views of the Hexagonal plate.
X YHP
VP
50°
40°
Locus of d2
a
b c
d
ef
a’ b’(f’) d’c’(e’)
30
a1’
b1’(f1’)
c1’(e1’)
d1’
a1
b1 c1
f1
e1
d1
a2’
b2’
f2’
c2’ e2’
d2’
a2
b2
c2
d2
e2
f2
EXAMPLE
Tutorial: VI
Projections of Solids (Axis perpendicular to one Plane)

• Components have different shapes.
• Any component will be formed of a single solid or combination of solids
of different shapes.
• The most common geometrical solids are prisms, cylinders, cones,
spheres, pyramids.
• Thus, drawing of components like machine parts require a thorough
knowledge of projections basic solids.
Basic building blocks
Projection of Solids
• A solid has three dimensions- length, breadth and thickness.
• At least two orthographic views are necessary to represent a solid.
• Sometimes, additional views projected on auxiliary planes become
necessary to make the complete description of a solid.

Types of Solids
1. Polyhedra
2. Solids of Revolution
Polyhedra:
• A polyhedron is defined as a solid bounded by planes called faces.
• When all the faces are equal and regular, the polyhedron is said to be regular.
• Seven regular polyhedra :
• Tetrahedron
• Cube or hexahedron
• Octahedron
• Dodecahedron
• Icosahedron
• Prism
• Pyramid

Tetrahedron:
• It has four equal faces, each an
equilateral triangle.
Cube or hexahedron
• It has six faces, all equal squares.
Octahedron
• It has eight equal equilateral triangles as faces.

Dodecahedron :
• It has twelve equal and regular pentagons as faces.
Icosahedron :
• It has twenty faces, all equal equilateral triangles.

Prism
• This is a polyhedron having two equal and similar faces called its ends or
bases, parallel to each other and joined by other faces which are
parallelograms.
• The imaginary line joining the centres of the bases is called the axis.
• A right and regular prism has its axis perpendicular to the bases.
• All its faces are equal rectangles.
Vertices (points)
Edges (lines)
Faces (planes)
6
9
5
The base has 3 sides.
Triangular Prism

Vertices (points)
Edges (lines)
Faces (planes)
8
12
6
Rectangular Prism
Vertices (points)
Edges (lines)
Faces (planes)
10
15
7
Pentagonal Prism

Vertices (points)
Edges (lines)
Faces (planes)
12
18
8
Hexagonal Prism
Vertices (points)
Edges (lines)
Faces (planes)
16
24
10
Octagonal Prism

Pyramid:
• This is a polyhedron having a plane figure as a base and a number of
triangular faces meeting at a point called the vertex or apex.
• The imaginary line joining the apex with the centre of the base is its axis.
• A right and regular pyramid has its axis perpendicular to the base which is a
regular plane figure.
• Its faces are all equal isosceles triangles.
Edges (lines)
4
6
3
Triangular Pyramid
Vertices (points)
Slant Faces (planes)
Apex or vertex

Edges (lines)
5
8
4
Rectangular Pyramid
Vertices (points)
Apex or vertex
Edges (lines)
6
10
5
Pentagonal Pyramid
Vertices (points)
Apex or vertex

Edges (lines)
7
12
6
Hexagonal Pyramid
Vertices (points)
Apex or vertex
Features of a Solid
Faces
Edge
Corner/Vertex
Axis
Axis
Invisible Edge
Base
Base
Lateral Face
Invisible Edge
Base edge

Apex or Vertex
Slant Face
Base edge
Base
Slant Edge
Axis
Features of a Solid
• Oblique prisms and pyramids
have their axes inclined to their
bases. (Not in syllabus)
• Prisms and pyramids are named
according to the shape of their
bases, as triangular, square,
pentagonal, hexagonal etc.
Solids of revolution:
• Cylinder
• Cone
• Sphere
Cylinder :
• A right circular cylinder is a solid generated by the revolution of a rectangle
about one of its sides which remains fixed.
• It has two equal circular bases.
• The line joining the centres of the bases is the axis. It is perpendicular to the
bases.

Cylinder :
Cone :
• A right circular cone is a solid generated by the revolution of a right-angled
triangle about one of its perpendicular sides which is fixed.
• It has one circular base.
• Its axis joins the apex with the centre of the base to which it is perpendicular.
• Straight lines drawn from the apex to the circumference of the base-circle are
all equal and are called generators of the cone.
• The length of the generator is the slant height of the cone.
***Oblique cylinders and cones have their axes inclined to their bases.

Cone :
Sphere :
• A sphere is a solid generated by the revolution of a semi-circle about its
diameter as the axis.
• The mid-point of the diameter is the centre of the sphere.
• All points on the surface of the sphere are equidistant from its centre.

• When a pyramid or a cone is cut by a plane parallel to its base, Thus
removing the top portion, the remaining portion is called its frustum
Frustum
Truncated Solids
• When a solid is cut by a plane inclined to the base it is said to be truncated.
• ***In our syllabus, mostly right and regular solids are prescribed. Hence, when
a solid is named without any qualification, it should be understood as being
right and regular.

Projections of Solids - Positions
A solid position may be defined by its axis position. Hence,
1. Solid axis perpendicular to the HP and parallel to VP.
2. Solid axis perpendicular to the VP and parallel to HP.
3. Solid axis parallel to both the HP and the VP.
4. Solid axis inclined to the VP and parallel to the HP.
5. Solid axis inclined to the HP and parallel to the VP.
6. Solid axis inclined to both HP and VP.
TUTORIAL VI
TUTORIAL VII
Not in
syllabus
Steps
1. Start the projection that shows the true shape and size of the base.
2. Identify the axis which is perpendicular to the any one reference plane.
3. On that reference plane draw the true shape.
4. When the axis is parallel to both the reference planes, the true shape of the
base can be seen from its side view where an auxiliary plane perpendicular
to both the reference planes can be used.
5. When the axis is inclined to any one reference plane, assume the axis is
perpendicular to that plane and draw the initial step, then proceed to
inclined position.

Axis perpendicular to the H.P.
Parallel to VP:
A
B
C
D
E
F
X HP
VP
Y
a’,(b’)
c’
d’,(e’) f’
b,(e)
a,(d)
c,(f)
Draw the projections of a triangular prism, base 40 mm
side and axis 50 mm long, resting on one of its bases on the
H.P. with a vertical face perpendicular to the V.P.
40
50
X HP
VP
Y
35°
40
40
a,(1)
c,(3)
b,(2)
d,(4)
a’ d’ b’ c’
1’ 4’ 2’ 3’
Draw the projections of the Cube
of side 40 nun resting on its base
on the HP and one of its vertical
faces inclined at 35° to the VP.
Tutorial: VI
Problem No. 1

X HP
VP
Y
45°
60
60a
b
b’a’ c’
o’
A Tetrahedron with a 60 mm
edge is resting on HP on one of
its face in such a way that one of
its face edges is inclined at 45°to
the VP. Draw is projections.
Tutorial: VI
Problem No. 2
o
c
X HP
VP
Y
30°
60
35
b’a’ c’ d’
A Pentagonal prism of base side
35 mm and axis length 60 mm
rests on the HP with it is base in
such a way that one of its base
edge is inclined at 30° to the VP.
Tutorial: VI
Problem No. 3
a,(1)
e’
2’1’ 4’5’ 3’
b,(2)
c,(3)
d,(4)
e,(5)

X HP
VP
Y
a
b c
d
A Hexagonal pyramid of base side 30
mm and axis length 60 mm rests on
the HP on its base. Draw its
projections when one of the base
edge is parallel to VP and the solid
axis is perpendicular to HP.
Tutorial: VI
Problem No. 4
30
o
a’
60
o’
b’(f’) d’
ef
c’(e’)
X HP
VP
Y
a
b
c
d
A Pentagonal pyramid of base side
30 mm and axis length 55 mm rests
on the HP on its base, in such a way
that one of its base edges is inclined
at 30° to VP and the solid axis is
perpendicular to HP. Draw its
projections
Tutorial: VI
Problem No. 5
30
o
a’
55
o’
e’b’ d’
e
c’
30°

X HP
VP
Y
40
a’,(1’)
a b cd
A Cube of side 40 mm is resting on
the HP on one of its edges. Draw its
projections when one of its faces
containing the resting edge is
inclined at 30° to the HP, and the
solid axis is perpendicular to VP.
Tutorial: VI
Problem No. 6
40
1
b’,(2’)
2
c’,(3’)
4 3
d’,(4’)
Axis perpendicular to the VP & Parallel to HP:
X HP
VP
Y
60
30
a’
d’
b,b’
a
c’
e’
e d c
o’
o
A Pentagonal pyramid of base side
30 mm and axis length 60 mm rests
on the HP on one of its base corner,
in such a way that one of the base
edges containing the resting corner
makes 40° to the HP. Draw its
projections when the axis is
perpendicular to the VP and the
base is touching the VP.
Tutorial: VI
Problem No. 7

A right Circular cone of base
diameter 30 mm and axis length 60
mm is resting on the HP on one of
its base circumferential point and
has its axis perpendicular to VP.
Draw its projections when its apex is
nearer to VP.
Tutorial: VI
Problem No. 8
X HP
VP
Y
a’
60
o’
b
φ30
a
b’
o
A Hexagonal prism of base side
25 mm and axis length 55 mm
lies on the ground on one of its
rectangular faces with its axis
perpendicular to VP. Draw its
projections when one of its end
faces is 15 mm infront of VP.
Tutorial: VI
Problem No. 9
X HP
VP
Y
15
35
a’,(1’)
a f,(b)
6,(2)
55
b’,(2’) c’,(3’)
d’,(4’)
e’,(5’)f’,(6’)
5,(3)1 4
e,(c) d

X HP
VP
Y
30°
25
X1
Y1
a’’(1”)
a’
55
a
b’’(2”)
c’’(3”)
d’’(4”)
e’’(5”)
b’ 2’
1’
e’
c’
d’ 4’
3’
5’
e
d
b
5
c
1
4
2
3
A Pentagonal prism of base
side 25 mm and axis length 55
mm lies on the HP on one of
its longer lateral edges with its
axis parallel to both HP and
VP. Draw its projections when
one of its rectangular faces
containing the resting edge is
inclined at 30° to the HP.
Tutorial: VI
Problem No. 10
Axis Parallel to HP & VP :
X HP
VP
Y
30°
30
X1
Y1
a’’
b’’
f’’
c’’
e’’
e’,(a’)
60
c’
d’,(b’)
f’
o’
o
c,(d)
A hexagonal pyramid of base
side 30 mm and axis length
60 mm lies on the HP on one
of its corners of the base with
its axis parallel to both HP
and VP. One of the base edges
containing the resting corner
is inclined at an angle of 30°
to HP. Draw its plan and
elevation.
Tutorial: VI
Problem No. 11
o’’
d’’
f,(c)
a,(b)

X HP
VP
Y
30°
30
X1
Y1
a’’
b’’
c’’
d’’
e’’
a’
60
b’
c’
d’
e’o’
o
e
a
b
c
d
A pentagonal pyramid of base
60 mm lies on the HP on one
of its corners of the base with
its axis parallel to both HP
and VP. One of the base edges
containing the resting corner
is inclined at an angle of 30°
to HP. Draw its plan and
elevation.
Tutorial: VI
Problem No. 12
o’’
Tutorial : VII
Projections of Solids (Axis Inclined to one Plane)
198TUTORIAL VII

PROJECTIONS OF SOLIDS WITH AXES INCLINED TO ONE OF
THE REFERENCE PLANES AND PARALLEL TO THE OTHER
In this case, the projections are drawn in two stages.
• Stage I : In the initial stage, the solid is assumed to be in simple position
• Stage II : Final projections may be obtained by one of the following two
methods:
• (i) Change of position method: The position of one of the views is altered as
required and the other view projected from it.
• (ii) Change of reference line (auxiliary plane) method : A new reference
line is drawn according to the required conditions, to represent an auxiliary
plane and the final view projected on it.
199TUTORIAL VII
A Triangular prism of
base side 35 mm and axis
length 55mm rests on the
HP on one of its base edge.
Draw its projections when
the solid axis is inclined at
35°to the HP and parallel
to VP.
1
Initial Position
Auxialiary Plane Method Change of position Method
200TUTORIAL VII

2
A Square prism of
base side 35 mm and
axis length 60 mm
rests on one of its
base edges on the HP
with its axis inclined at
30° to the HP and
parallel to the VP.
Draw its top and front
views.
Initial Position
Auxialiary Plane Method
Change of position Method
201TUTORIAL VII
3
A rectangular prism 50
x 25 mm base and
length 70 mm rests
with one of its longer
edges of the base on
HP, and the axis is
inclined at 30° to HP
and parallel to VP.
Initial Position Auxialiary Plane Method
202TUTORIAL VII

4
A Pentagonal prism of
base side 25 mm and
axis length 55 mm rests
on the HP on one of its
base edges. Draw its
projections when the
solid axis is inclined at
45° to the HP and
parallel to the VP. Draw
the projections of the
prism by auxiliary
position method.
Initial Position
203TUTORIAL VII
5
A Hexagonal prism of
base side 30 mm and
axis length 40 mm is
resting on the HP on one
of its base edge. Draw
its projections when the
base containing the
resting edge is inclined
at 60° to the HP and the
solid axis is parallel to
VP.
Initial Position
204TUTORIAL VII

6
A square pyramid of base
50 mm rests on the HP on
one of its base edge. Draw
40°to HP and parallel to
VP.
Initial Position
205TUTORIAL VII
A Pentagonal
pyramid of base edge 30
mm and axis length 60
mm rests on one of its
base edges on the HP
with its axis inclined at
30° to the HP and
parallel to the VP. Draw
its top and front views.
7
Initial Position
206TUTORIAL VII

8
edge 25 mm and axis length
60 mm rest on the base side
on HP such that the highest
base corner is 20 mm above
HP. Its axis is parallel to the
VP. Draw its top and front
views.
Initial Position
207TUTORIAL VII
9
A hexagonal pyramid of
base side 25 mm and
on the HP on one its
base edge. The solid
axis is inclined at 40°to
HP and parallel to VP.
Initial Position
208TUTORIAL VII

10
Draw the projections of a square
pyramid of base side 25 mm and
altitude 50 mm when its rests on the
ground on one of its base edges
perpendicular to the VP and the
triangular face containing the resting
edge is perpendicular to the VP and
the HP.
Initial Position
209TUTORIAL VII
11
A Pentagonal pyramid of
base side 30 mm and
altitude 75 mm rests on
the HP on one of its
base edges such that
the triangular face
containing the resting
edge is perpendicular to
both HP and the VP.
Initial Position
Change of position Method210TUTORIAL VII

12
A Hexagonal pyramid
of base side 30 mm and
altitude 75 mm rests
base edges such that
the triangular face
edge is perpendicular
to both HP and the VP.
Initial Position
211TUTORIAL VII
13
Draw the projections of the
square pyramid of base side
60mm and altitude 100mm
when it lies on the ground
on one of its triangular
faces / slant faces and the
solid axis is parallel to the
VP.
Initial Position
212TUTORIAL VII

14
A Pentagonal pyramid
of base side 30 mm
and axis length 55
mm rests on the
ground on one of its
lateral surface with
its solid axis parallel
to VP. Draw its
projections.
Initial Position
213TUTORIAL VII
15
A Hexagonal pyramid
of base side 30 mm
and axis length 60
mm rests on the HP
on one of its
triangular faces with
its axis parallel to VP.
Initial Position
214TUTORIAL VII

16
Draw the projections of
a Pentagonal prism of 25
mm side and axis length
50 mm resting on the HP
on one of its corner in
such a way that the two
base edges passing
through it makes equal
inclinations with HP.
when the solid axis is
inclined at 30° to the
ground and parallel to
VP.
Initial Position
215TUTORIAL VII
17
Cube of side 40 mm when it
rests on the ground on one
of its corners with the
solid diagonal vertical.
Initial Position
216TUTORIAL VII

18
Hexagonal prism of 25
mm side and axis length
50 mm resting on the HP
on one of its corner in
such a way that the two
base edges passing
through it makes equal
inclinations with HP. Draw
30° to the ground and
parallel to VP.
Initial Position Auxialiary Plane Method Change of position Method
217TUTORIAL VII
19
A hexagonal prism of
base side 25mm and
axis length 55 mm
rest on the HP on one
of its base corners
such that the solid
diagonal passing
through that corner is
perpendicular to the
HP. Draw its
projections.
Initial Position
218TUTORIAL VII

20
A Cylinder of base
diameter 50 mm and
resting on the HP on a
base circumferential
point so that its axis
is inclined at 45° to
the HP and parallel to
the VP. Draw its plan
and elevation.
Initial Position
219TUTORIAL VII
21
A cylinder of diameter
35 mm and axis length
55 mm is resting on
the ground on its base.
It is then tilted such
that a solid diagonal is
vertical. Draw its
projections.
Note: a’ v’ is considered as solid diagonal
Initial Position
220TUTORIAL VII

22
A square pyramid of
base side 40 mm and
base corners in such a
way that the base edges
corner is equally
inclined to the HP. Draw
40° to the HP and
parallel to VP.
Initial Position
Auxialiary Plane Method Change of position Method221TUTORIAL VII
23
of base edge 30 mm
and axis length 60 mm
rests on one of its
base corners on the HP
in such a way that the
two base edges
corner is equally
inclined to HP. Draw
its projections with its
solid axis inclined at
30° to the HP and
parallel to the VP.
Draw its top and front
views.
Initial Position
222TUTORIAL VII

24
hexagonal pyramid of
base side 25 mm and
axis length 55 mm
resting on the HP on
one of its base corner.
The axis of the solid is
HP and parallel to VP.
Initial Position
223TUTORIAL VII
25
A Cone of base
diameter 50 mm
and altitude 70
mm rests on the
HP on a base
circumferential
point with its
axis is inclined at
30° to the HP
and parallel to
the VP. Draw its
front and top
views.
Initial Position
224TUTORIAL VII

26
A square pyramid of
base side 40 mm and
axis length 65 mm
rests on the HP on one
of its base corners in
such a way that the
slant edge containing
the resting corner is
perpendicular to HP
and parallel to VP.
Draw the projections.
Initial Position
Auxialiary Plane Method Change of position Method 225TUTORIAL VII
27
base edge 30 mm and axis
length 60 mm rests on
one of its base corners on
the HP in such a way that
the two base edges
corner is equally inclined
to HP. Draw its
slant edge containing the
resting edge is
perpendicular to HP and
parallel to VP.
Initial Position
226TUTORIAL VII

28
Draw the projections
of hexagonal pyramid
of base side 25 mm and
axis length 55 mm
resting on the HP on
one of its base corner
and the slant edge
corner is perpendicular
to HP and parallel to
VP.
Initial Position
227TUTORIAL VII
29
A cone of base diameter 50 mm
and axis length 60 mm has one
of its base circumferential
points on HP. Draw its
projections when one of the
generators of the cone is
perpendicular to HP and parallel
to VP.
Initial Position
228TUTORIAL VII

30
A square pyramid of
base side 40 mm and
axis length 65 mm
rests on the HP on
one of its slant
edges. Draw its
solid axis is parallel
to VP.
Initial Position
229TUTORIAL VII
31
length 55 mm rests on the
ground on one of its
lateral edges such that
the triangular faces
edge is equally inclined to
HP. Draw its projections
parallel to VP.
Initial Position
230TUTORIAL VII

32
A Hexagonal pyramid of
base side 30 mm and
slant edges in such a way
that the triangular face
edge is equally inclined
to HP and the solid axis
parallel to VP. Draw its
projections.
231TUTORIAL VII
33
A Cone of base
diameter 50 mm and
axis length 60mm has
one of its generators
on HP with the solid
axis parallel to VP.
Initial Position
232TUTORIAL VII

34
base side 25 mm and
lying on the HP on one
of its rectangular
faces with its axis
VP. Draw its
projections.
Initial Position
233TUTORIAL VII
35
A Pentagonal prism
of base side 25 mm
and axis length 55
mm is resting on HP
on one of its
rectangular faces
with its axis
inclined at 45°to
the VP. Draw its
projections.
Initial Position
234TUTORIAL VII

36
A Square pyramid of
base side 30 mm and
axis length 50 mm
its projections when
the solid axis is
inclined at 45° to VP
and parallel to HP.
Initial Position
235TUTORIAL VII
37
A pentagonal pyramid
of base side 25 mm
and axis length 55 mm
the solid axis is
and parallel to HP
Initial Position
236TUTORIAL VII

38
base side 25 mm and
axis length 55 mm
the solid axis is
and parallel to HP.
Initial Position
237TUTORIAL VII
39
A square prism of
base side 35 mm
and axis length 60
mm lies on the HP
on one of its longer
edges with its
rectangular faces
equally inclined to
the HP. Draw its
projections when
its axis is inclined
at 30° to the VP.
Initial Position
238TUTORIAL VII

40
length 55 mm lies on the
HP on one of its lateral
edges in such a way that
one of its rectangular
face is perpendicular to
HP. Draw its projections
inclined at 40° to the VP
and parallel to the HP
Initial Position
239TUTORIAL VII
41
base side 25 mm and
axis length 55 mm lies
lateral edges in such a
way that the
rectangular faces
containing the lateral
edge is equally inclined
to HP. Draw its
35° to VP and parallel
to the HP.
Initial Position
240TUTORIAL VII

42
side of base 25 mm
and axis 55 mm long
is resting on a lateral
edge on HP. The
rectangular face
containing that edge
is inclined at 30° to
the HP and the solid
axis is inclined at 40°
to the VP. Draw its
projections.
Initial Position
241TUTORIAL VII
43
a hexagonal prism of
base side 20 mm and
axis length 50 mm when
it is lying on the ground
on one of its lateral
edge, in such a way that
the two rectangular
faces containing the
resting edge is equally
inclined to HP. Draw its
axis is inclined at 35°to
the VP and parallel to
the HP.
Initial Position
242TUTORIAL VII

44
A Cylinder of base
diameter 50 mm and
axis length 60 mm
rests on the HP on
one of its generators.
when the cylinder
axis is inclined at 30°
to the VP and parallel
to HP.
Initial Position
243TUTORIAL VII
45
A Square pyramid of
the HP on one of its base
corners with a base side
corner makes an angle of
35°with the HP. Draw its
projection when the solid
axis is inclined at 30° to
the VP and the vertex is
away from the VP.
Initial Position
244TUTORIAL VII

46
base side 25mm and
axis length 55 mm
of its base corner in
such a way that two of
its base edge is equally
inclined to the HP.
VP and parallel to HP.
Initial Position
245TUTORIAL VII
47
side 30 mm and axis
edge and has one of its
lateral surface on VP
with the solid axis
parallel to the HP. Draw
its projections.
Initial Position
246TUTORIAL VII

48
the VP on one of its
lateral surface with its
solid axis parallel to HP.
Initial Position
247TUTORIAL VII
49
the VP on one of its
triangular faces with its
axis parallel to HP. Draw
its projections.
Initial Position
248TUTORIAL VII

50
side 30 mm and axis
corner and has one of its
lateral edge on VP with
the solid axis parallel to
the HP. Draw its
projections.
Initial Position
249TUTORIAL VII
51
A cone of base diameter
50 mm and axis length 60
mm has one of its
generators on VP with the
solid axis parallel to and 40
mm above the HP. Draw its
projections.
Initial Position
250TUTORIAL VII

52
60 mm has one of its lateral
edges on VP and the solid
axis is parallel to HP.
Initial Position
251TUTORIAL VII
53
base side 30 mm and
axis length 60 mm has
one of its lateral edge
on VP and the solid axis
is parallel to HP. Draw
its projections.
Initial Position
252TUTORIAL VII

54
A Tetrahedron of edges 50
mm rest on one of its edges
on the VP. One of the faces
containing the resting edge
is inclined at 30° to the VP.
Draw its projections of the
tetrahedron when the
resting edge is normal to
the HP.
Initial Position
253TUTORIAL VII
55
A square pyramid of
base side 30 mm and
suspended by means
of a string from one
of its base corners
with its axis parallel
to VP. Draw its
projections.
Initial Position
Note: The pyramid is suspended from the corner “A”
254
TUTORIAL VII

56
of base side 25 mm
and 60 mm height is
suspended from a
string attached to
one corner of the
base of the pyramid.
The string is held,
such that the apex of
the pyramid is just
touching HP. The axis
of the pyramid is
parallel to VP. Draw
the projections of
the pyramid.
Initial Position
Note: The pyramid is suspended from the corner “D”
255
TUTORIAL VII
57
A Hexagonal pyramid
of base side 30 mm
and axis length 60
mm is suspended by
means of a string
from one of its base
corners with its axis
its projections.
Initial Position
Note: The pyramid is suspended from the corner “A”
256
TUTORIAL VII

58
base side 25 mm and
suspended freely by
means of a string
from one of its base
corners with its axis
its projections.
Initial Position
Note: The prism is suspended from the corner “A”
257
TUTORIAL VII
Tutorial: VIII
Sections of Solids
Staff Incharge : A.KUMARASWAMY

Representation of Section Plane

Types of Sectional Planes and Sectional Views
Classification of Section Planes
Plane Perpendicular to VP and Parallel to HP
Plane Perpendicular to HP and Parallel to VP
Plane Perpendicular to VP and inclined to HP
Plane Perpendicular to HP and inclined to VP
Plane Perpendicular to both VP and HP
Plane inclined to HP both VP and HP

Sectioning is a technique by which the object is sliced and the cutaway view of the part is then drawn.
To show the construction clearly in many hollow objects, the front part is imagined to be removed to
reveal the interior surface.
Where and when this occurs the edges are represented with solid lines and the cut surface is
crosshatched or section lined.
The interior detail is now shown more clearly because the hidden lines have been replaced with visible
object lines.
The object cut Section technique
Section Views

Hatching
1
A Cube of side 40 mm rests with one of its
square faces on HP such that one of its
vertical square faces is inclined at 30° to VP. A
section plane, parallel to VP and perpendicular
to HP passes through the solid at a distance of
10 mm away from its vertical axis and infront
of it. Draw its top view, sectional front view.

2
A Hexagonal pyramid of base side 30 mm and axis
length 60 mm rests on the HP on its base such
that one of its base edges is parallel to VP and the
solid axis is perpendicular to VP. A cutting plane
parallel to VP and perpendicular to HP cuts the
solid at a distance of 15 mm away from the axis.
Draw its top view, sectional front view.
3
A Pentagonal pyramid of base side 30 mm and
axis length 60 mm rests on the HP on its base
in such a way that one of its base edges is
parallel to VP and the solid axis perpendicular
to HP. A section plane parallel to the ground
and perpendicular to VP cuts the solid axis at a
distance of 20 mm away from the vertex. Draw
its front view and sectional top view.

4
A square prism of base side 30 mm and height
60 mm rests on the HP on one of its ends with
two of its rectangular faces equally inclined to
the VP. It is cut by a plane perpendicular to the
VP and inclined at 60° to the HP meeting the
axis at 15 mm from the top. Draw its front
view, sectional top view and true shape of the
section.
Sectional top view
Front view
True shape
5
A Pentagonal prism of base side 40 mm and axis length 75
mm rests on the HP on one of its ends with a rectangular
face parallel to the VP. It is cut by a plane perpendicular to
the VP, inclined at 300 to the HP and meeting the solid axis
at 25 mm from the top face. Draw the front, sectional top
view and true shape of the section.

6
A hexagonal prism of base side 25 mm and axis
length 50 mm rests on the HP on its base .in
such a way that one of its rectangular face is
parallel to VP. It is cut by a plane inclined at
50° to the base and bisecting the solid axis.
Draw the front view, sectional top view and
true shape of the section..
Sectional top view
True shape
7
A square pyramid of base 30 mm and axis
60 mm long is standing on the HP with its
base edge equally inclined to VP. It is cut
by a section plane perpendicular to the VP
and inclined at 30° to the HP, bisecting the
axis. Draw the sectional top view and the
true shape of the section, if the upper
portion is removed.
Sectional top view
True shape

8
A rectangular pyramid of base 30 mm X 50
mm and axis 50 mm is resting on its base
with the longer edge of the base parallel to
VP. It is cut by a section plane perpendicular
to the VP, inclined at 30° to HP and passing
through a point on the axis 20 mm from the
apex. Draw the front view, the sectional top
view and the true shape of such a section of
the pyramid
Sectional top view
Front view
True shape
9
A pentagonal pyramid, side of base 30 mm
and height 52 mm, stands with its base on
HP and an edge of the base is parallel to
VP and nearer to it. It is cut by a plane
perpendicular to VP, inclined at 40° to HP
and passing through a point on the axis, 32
mm above the base. Draw the sectional top
view and true shape of the section.
Sectional top view
Front view
True shape

10
A hexagonal pyramid of base side 30 mm
and height 60mm rests vertically on its
base on the ground with two of its base
sides parallel to VP. Is is cut by a sectional
plane inclined at 30° to HP and
perpendicular to YP and meeting the axis
at the midpoint. Draw the front view, the
sectional top view and the true shape of
such a section of the pyramid
Sectional top view
Front view
True shape
11
A hexagonal pyramid of base side 25 mm and axis
55mm length rests on its base on the HP with two
base edges perpendicular to the VP. It is cut by a
plane perpendicular to the VP and inclined at 30° to
the HP meeting the axis at 20 mm from the vertex.
Draw the front view, sectional top view and the true
shape of the section.
Sectional top view
Front view
True shape

12
A cylinder of diameter 50 mm and height
60 mm rests on its base on HP. It is cut by
a plane perpendicular to VP and inclined at
45°to HP. The cutting plane meets the axis
at a distance of 15 mm from the top face.
Draw the sectional plan and true shape of
the section.
Sectional top view
Front view
True shape
13
A cone of base diameter 65 mm and axis 80
mm long is resting on its base on HP. It is cut
by a section plane perpendicular to VP and
inclined at 45° to HP. The cutting plane passes
through the axis at a distance of 40 mm above
HP. Draw the sectional top view and true shape
of the section.
Sectional top view
Front view
True shape

14
A Cone of base diameter 50mm and
altitude 60mm rests on its base on the HP.
It is cut by a plane perpendicular to the VP
and parallel to one of the extreme
generators, 10 mm away from it. Draw the
sectional top view and true shape of the
section.
y
15
A Cone of base diameter 40 mm and altitude
50 mm rests on its base on HP. It is cut by a
section plane perpendicular to VP and
inclined at 80o to HP, passing through the
apex. Draw the front view, sectional top
view and true shape of the section

16
A hexagonal prism of base side 25 mm and height
50 mm rests on the HP on one of its ends with two
rectangular faces parallel to the VP. It is cut by a
plane perpendicular to the HP and inclined at 50° to
the VP at a distance of 10mm away from the axis.
Draw the .top view, sectional front view and true
shape of the section.
Sectional front view
Top view
True shape
17
A pentagonal pyramid of base side 25 mm and axis
length 60 mm rests on the HP on its base with an
edge of the base parallel to the VP. It is cut by a
vertical plane inclined at 45° to the VP at a
distance of 8 mm away from the axis. Draw the top
view, sectional front view and true shape of the
section.
Top view
True shape

18
A hexagonal pyramid of base side 25mm and axis 55
mm rests on HP on its base with two edges parallel to
VP. It is cut by a vertical plane inclined at 30° to VP
and cutting the pyramid at a distance of 6 mm away
from the plan of the axis. Draw the top view, sectional
front view and true shape of the section..
Top view
True shape
19
A cylinder of base diameter 60mm and axis height
65mm is resting on the ground on its base. It is cut
by a plane perpendicular to HP, inclined at 30° to
VP that passes through a point at a distance of 15
mm away from the axis. Draw the sectional
elevation and true shape of the section
A.KUMARASWAMY. AP, MECH Sectional front view
Top view
True shape

20
A vertical cone of diameter 60 mm, height 80 mm
is cut by a cutting plane perpendicular to HP and is
inclined at 40° to VP, passing through a point at a
distance of 15 mm away from the axis. Draw the
sectional elevation and true shape of the section
Top view
True shape
21
A cone of diameter 55 mm and axis
height 70 mm is resting on the
ground on its base. A section plane
perpendicular to both HP and VP
cuts the solid at a distance of 7 mm
away from the axis. Draw the
sectional elevation and true shape of
the section.
Front view
Top view
True shape

22
A square pyramid of base side 35 mm and
axis length 60 mm is resting the HP on its
base with a side of base inclined at 45° to
the VP. It is cut by a plane perpendicular to
both HP and VP meeting at a distance of
10mm away from the axis. Draw its top
view, front view and true shape of the
section.
Front view
Top view
True shape
23
A cube of side 50 mm is placed and cut by a plane
in such a way that the true shape of the section is
regular hexagon. Draw the front and top views of
the cube and find the inclination of the cutting
plane with the HP.

24
A Tetrahedron of edge 60 mm rests on the HP on
one of its face such that one of the edges of the
resting face is perpendicular to VP. It is cut by a
plane perpendicular to VP and inclined to the HP in
such a way that the true shape of the section is
an isosceles triangle of base 40 mm and altitude
30 mm. Find the inclination of the cutting plane
with the HP. Also draw the front view, sectional
top view and true shape of the section
25
A tetrahedron of edge 30 mm rests on the HP
on one of its faces. It is cut by a plane
perpendicular to the VP and inclined to the HP
such that the true shape of the section is a
square. Draw the front view showing the
cutting plane, sectional top view and the true
shape of the section
Front view
Sectional top view
True shape

26
A square pyramid of base side 50 mm and height
65 mm is resting on HP with a base edge parallel
to VP. It is cut by a plane perpendicular to VP,
inclined to HP in such a way that the true shape is
a trapezium of sides 40 mm and 20 mm. Draw the
projection and the find the angle of the cutting
plane. Also draw the true shape of the section
Front view
Sectional top view
True shape
27
A cone of base diameter 40 mm and altitude 50
mm rests on its base on HP. It is cut by a
section plane perpendicular to VP and inclined
to HP in such a way that the true shape of the
section is an isosceles triangle of base 25 mm.
Draw the front view, sectional top view and
true shape of the section
Front view
Sectional top view
True shape

28
A Sphere of diameter 80 mm is cut by a HT inclined at
40o to VP. The cutting plane is located at a minimum
distance of 20 mm from the center of the sphere.
Draw its top view, sectional front view and true shape
of the section.
29
A cone of base diameter 40 mm and height
50 mm rests on its base on the HP. It is
cut by a plane perpendicular to the VP and
inclined at 400 to the HP. The cutting
plane meets the axis at 20 mm from the
vertex. Draw the sectional top view and
the true shape of the section.

30
A Hexagonal pyramid of base side 25 mm and
axis 55 mm rests on HP on its base with two
edges parallel to VP. It is cut by a vertical
plane inclined at 30o to VP and cutting the
pyramid at a distance of 6 mm away from the
plan of the axis. Draw the top view, sectional
front view and true shape of the section.
31
A Cylinder of diameter 50mm and height 65 mm
rests on its base on the HP. It is cut by a plane
Perpendicular to the VP and inclined at 600 to the
HP. The cutting plane meets the axis at a
distance of 40 mm above the base. Draw the
sectional top view and the true shape of the
section.

32
A Pentagonal prism of base side 25 mm and
axis length 55 mm rests on the HP on its
base such that one of its base edges is
perpendicular to VP and the solid axis is
perpendicular to HP. A section plane
parallel to the base and perpendicular to VP
cuts the solid axis at a distance of 15 mm
away from the top face. Draw its front
view and sectional top view.
Tutorial: IX
DEVELOPMENT OF SURFACES
Staff Incharge : A.KUMARASWAMY

Development of a different surfaces
Methods of Development

Steps involved in Development of a solid
πD
H
D
Where H= Height D= base diameter
Development of lateral surfaces of solids.
Cylinder: A Rectangle
Note: Lateral surface is the surface excluding top & base
Complete Development

Tetrahedron: Four Equilateral Triangles
Cube: Six Squares.
S
H
H= Height S = Edge of base
Prisms: No. of Rectangles
S
H

θ
θθθθ = R
L
3600
R=Base circle radius.
L=Slant height.
Cone: Sector of circle
L
L
L= Slant edge. S = Edge of base
Pyramids: (No.of triangles)
L
s
L
s

1
A Rectangular prism of cross section 45 X 30 mm and height 55 mm rests
on the ground on one of its ends with one of its longer edges of the base
inclined at 300 to the VP. It is cut by a plane perpendicular to VP and
inclined at 300 to the HP. The cutting plane meets the axis at a point 45
mm above the base. Draw the development of the surface of the lower
part of the prism.
2
A Hexagonal prism of side 30 mm and axis length 60 mm is resting
on HP on its base with two of its vertical faces perpendicular to VP.
It is cut by a plane inclined at 500 to HP and perpendicular to VP
and meets the axis of prism at a distance of 10mm from the top
end. Draw the development of the lateral surface of the prism.

3 A Pentagonal prism of base side 30mm and axis length 60mm is resting on HP on its base with a side of
base parallel to VP. It is cut by a plane inclined at 350 to HP and perpendicular to VP and meets the axis at
a distance of 35 mm from the base. Draw the development of the lower portion of the prism.
4
A Cylinder of base diameter 50 mm and axis length 65 mm resting
on HP on its base is cut by a plane inclined at 600 to HP and
perpendicular to VP. The cutting plane passes through a point on
the axis at a distance of 25 mm from the top end. Draw the
development of the lateral surface of the right portion of the
cylinder

5 A Cylinder of diameter 50 mm and axis height 65 mm is cut by a plane perpendicular to VP and inclined at
60° to the HP into two equal parts. Draw the development of the lateral surface of the truncated solid.
6 A rectangular pyramid 60 mm x 50 mm and height 75 mm is
resting on its base on HP with its longer base edges parallel to
VP. It is sectioned by a plane perpendicular to VP, inclined at 65°
to HP and passing through the mid-point of the axis. Develop the
lateral surfaces of the cut pyramid

7 A Hexagonal pyramid of base side 25 mm and altitude 50 mm is
resting vertically on its base on the ground with two of the side
of the base perpendicular to the VP. It is cut by a plane
perpendicular to VP and inclined at 400 to the HP. The cutting
plane bisects the axis of the pyramid. Draw the development of
the lateral surfaces of the pyramid.
8
A regular pentagonal pyramid of side 40 mm and altitude 75 mm has its base
on HP with a side of base perpendicular to VP. The pyramid is cut by a section
plane perpendicular to VP and inclined at 300 to the HP. The cutting plane
meets the axis of the pyramid at a point 30 mm below the vertex. Obtain the
development of the remaining part of the pyramid.
30

9
A right circular cone of base diameter 60 mm and height 70 mm is resting on its base on the
ground. It is cut by a plane inclined at 300 to HP and perpendicular to VP. The cutting plane
bisects the axis of the cone. Draw the development of the lateral surface of the truncated
cone.
9

10
A Cone of base diameter 60 mm and height 70 mm is resting on its base on the ground. It is
cut by a plane perpendicular to the VP and parallel to the HP at a distance of 20 mm from
the vertex. It is also cut by a plane inclined at 400 to the base and meeting the axis at a
point 20 mm above the base. Draw the development of the lateral surface of the cut cone.
10

11
A Pentagonal prism of base side 25 mm and height 60 mm stands on
one of its ends on the HP with a rectangular face parallel to the VP.
A hole of diameter 30 mm is drilled centrally through the prism in
such a way that the axis of the hole bisects the axis of the prism at
right angles. The axis of the hole is perpendicular to the VP. Draw
the development of the lateral surfaces of the prism.
12 A Square pyramid of base side 35 mm and axis 60 mm rests on its
base on the HP with one of the sides of the base inclined at 300 to
the VP. A string is wound round the surfaces of the pyramid starting
from left extreme point on the base and ending at the same point.
Find the shortest length of the string required. Also trace the path
of the string in the front and top views.

13 A cone of base diameter 60 mm and height 70 mm rests vertically on
its base on the ground. A string is wound round the curved surface
of the cone starting from the left extreme point on the base and
ending at the same point. Find the shortest length of the string
required. Also, trace the path of the string in the front and the top
views.
14
A Cone of base diameter 60 mm and axis height 70 mm is
resting on the HP on its base. It is cut by a plane
perpendicular to both HP and VP, 10 mm away from the axis.
Draw the development of lateral surfaces

15
A right regular Hexagonal pyramid of 30 mm side of base and height
70mm stands with its base on HP. A through circular hole of 30 mm
diameter is drilled through the pyramid such that the axis of the
hole is perpendicular to VP and intersects the axis of the pyramid at
a distance of 20 mm above the base. Draw the development of the
lateral surface of the pyramid showing the true shapes of the holes
formed on it.
16
A Funnel tapers from a circular opening of diameter 70 mm to a circular opening of
diameter 20 mm over an axial length of 50 mm and extends axially to a further
distance of 40 mm. There is a cylindrical portion of height 15 mm above the
tapering portion. Develop the funnel.

16
Tutorial : X
Isometric Projection
Staff Incharge: A.Kumaraswamy

Pictorial Projections
(i) Axonometric projection.
A. Isometric projection.
B. Dimetric projection.
C. Trimetric projection.
(ii) Oblique projection
(iii) Perspective projection
Isometric Projection
• It is a method of producing pictorial view of an object showing all
three faces of the object simultaneously on a single reference plane.
• It is a type of parallel projection
• The angles between any two of the three coordinate axes is 120°

• Observer at infinity
• Projectors parallel to each
other and perpendicular to
projection plane
• Object is inclined with respect
to projection plane
Isometric Terminology
• The three coordinate axes are called isometric axes
• Any line parallel to isometric axes is called isometric line
• A non-isometric line is a line not parallel to any one of the three
isometric axis
• In isometric projection of cube, the faces of the cube and any plane
parallel to them is called isometric planes

Principle of isometric projection
Isometric Scale Isometric Length= 0.82 x True length

Note: ISOMETRIC DRAWING = ISOMETRIC VIEW
Difference Between Isometric view and Isometric Projection
Difference Between Isometric view and Isometric Projection

DIMENSIONING IN ISOMETRIC PROJECTION
Note: Dimensioning is same as in orthographic projections, but
dimensioning lines should be parallel to isometric lines.
ISOMETRIC PROJECTION
PROBLEM:-
Draw the isometric views of the following: (i) A square with its surface vertical and horizontal
(ii) A rectangle with its surface vertical and horizontal
(i)
(ii)

PROBLEM:-
Draw the isometric view of the triangle shown in below
PQNM – helping rectangle
PROBLEM:-
Draw the isometric view of a hexagon with its surface vertical and horizontal.
Helping Diagram

PROBLEM:-
Draw the isometric views of a circle
Points Method
PROBLEM:-
Draw the isometric views of a circle
Four Centre Method

PROBLEM:-
Draw the isometric view of the semicircle
PROBLEM:-
Draw the isometric view of the plane figure shown in Figure below

PROBLEM:-
Draw the isometric view of a square prism of base side 20 mm and height 30 mm
PROBLEM:-
Draw the isometric view of a
cylinder resting on one of its
ends on the HP.

PROBLEM:-
Draw the isometric
view of a hexagonal
pyramid of base of
side 15 mm and
height 55 mm
resting on its base
on the lIP with two
sides of the base
parallel to the VP
PROBLEM:-
Draw the isometric
view of a right
circular cone of
base diameter 34
mm and height 60
mm when it is
resting on its base
on the HP.

1
Draw the isometric view of
(a) Hexagonal prism (b)
Pentagonal prism of base
side 25 mm and axis height
50 mm when it rests on one
of its ends on the HP with
one of its base sides
parallel to VP.
1 a
1 b

2a
Draw the isometric view of
(a) Cylinder (b) Cone of base
diameter 60 mm and axis
length 70mm when it’s
resting on its base on HP.
2b

3a
Draw the isometric
projection of (a)
Hexagonal Pyramid (b)
Pentagonal pyramid of
base side 30 mm and
axis length 60mm when
it rests on its base
with one of its base
side perpendicular to
VP.
3
3b

4
base edge 25 mm and
height 50 mm rests on
the HP on its base with
two of its rectangular
face perpendicular to the
VP. It is cut by a plane
inclined at 300 to HP and
perpendicular to VP at a
distance of 40 mm from
its base. Draw the
isometric view of the
truncated prism.
5
A Pentagonal prism of base side 35 mm and axis length 70 mm rests on the
ground with its two adjacent rectangular faces equally inclined to VP and
nearer to the observer. A section plane perpendicular to the VP and inclined at
450 to the HP passes through a point on the axis 50 mm above the base of the
prism. Draw the isometric projection of the truncated prism.

6
base side 30mm and axis
length 65mm is resting on
HP on its base with a side
of base perpendicular to
VP. It is cut by a plane
inclined at 300 to HP and
perpendicular to VP and
passes through a point at a
distance of 30 mm from
the apex. Draw the
remaining portion of the
pyramid.
7
Draw the isometric view
of a frustum of a
Hexagonal pyramid when
it is resting on its base
on the HP with two
sides of the base
parallel to the VP. The
side of base is 30 mm
and top face is 10 mm.
The height of the
frustum is 55 mm.

8
A Cylinder of base
diameter 60 mm and
height 70 mm rests with
its base on HP. A section
plane perpendicular to VP
and inclined at 450 to HP
passes through the axis
at a distance of 50 mm
above its base. Draw the
truncated cylinder
showing the cut surface..
9
A Cone of base diameter 60 mm and height 65 mm rests with its base
on HP. A cutting pane perpendicular to VP and inclined at 300 to HP cuts
the cone such that it passes through a point on the axis at a distance of
30 mm above the base of the cone. Draw the isometric view of the
truncated cone showing the cut surface.

10
A combination of the solids is formed as follows: A frustum of a Cone
25 mm top diameter, 50 mm bottom diameter and 50 mm height is
placed vertically on a Cylindrical block of 75 mm diameter and 25 mm
thick such that both the solids have the common axis. Draw the
isometric view of the combinations of the solid.
11
Three square rods of 30mm x 30mm cross section and lengths
100mm, 100mm and 60mm are so nailed together that they form
the letter ‘H’. Draw the isometric view of the letter ‘H’

12
A dust bin is in the form of a frustum of a hollow
Square pyramid with the base dimensions of 20 mm
sides and the top open surface of 45 mm sides. Draw
the isometric view of the hollow dust bin, if its height
is 50 mm and the wall thickness is negligible.
13
A cylinder of base diameter 30 mm and axis 50 mm is placed on its
base centrally on the top of a square slab of side 50 mm and
thickness 20 mm. Draw the isometric projection of the combination
of solids.

14
A hemispherical vessel of diameter 90 mm is placed centrally over a
cylinder of diameter 60 mm and height 75 mm which in turn is kept
centrally over a square prism of base side 80 mm and height 20mm.
Draw the isometric projection of the disposition of the solids.
Tutorial : XI
Perspective projection

• It is a method of producing pictorial view of an object showing all
three faces of the object simultaneously.
• It is a type of photographic projection
Horizon Plane
Central Plane
Auxiliary Ground Plane
Ground Plane
III Quadrant
Station Point
Axis of Vision

1
A Square prism of base 25 X 25 mm and height
40 mm rests on the GP with the edges of the
base making 450 with the PP. The vertical edge
nearer to the PP is 25 mm to the right of the
station point and 25 mm behind the PP. The
station point is 55mm above GP and 70 mm in
front of PP. Draw the perspective view of the
solid.
2
A Rectangular prism 40X30X15 mm rests on the
ground on one of its ends with one of the longest
edges touching the PP and the shortest edges
receding to the left at an angle of 400 to the PP.
The nearest vertical edge is 15 mm to the left of
the station point, which is at a distance of 55mm
in front of the PP and 30 mm above the ground.
Draw the perspective view of the solid.

3
A Hexagonal prism of base side 25 mm and
height 50 mm is resting with its base on the GP
such that one of its rectangular faces is inclined
at 300 to the PP and the vertical edge nearer to
PP is 15 mm behind it. The station point is 45 mm
in front of the PP, 70 mm above the GP and lies in
a central plane, which is 15 mm to the left of the
vertical edge nearer to the PP. Draw the
perspective view of the solid.
4
A Square pyramid of base side 30 mm and
altitude 40 mm rests on its base on the ground
such that one of its base sides is parallel to the
picture plane and 10 mm in front of it. The
station point is 50 mm in front of the picture
plane, 25 mm to the left of the axis of the
pyramid and 55 mm above the ground. Draw the
perspective view of the pyramid.

5
A Hexagonal pyramid of base side 25 mm and
axis length 50 mm is resting on the GP on its base
with a side of base inclined at 300 to PP. The
nearest corner to the PP is 20mm behind it. The
station point is 60mm above the GP and 80 mm in
front of PP and lies in a central plane, which is 50
mm to the left of the axis of the pyramid.
6
A frustum of a Square pyramid of base edge 26
mm, top edge 20 mm and height 35 mm rests on
its base on the ground with base edges equally
inclined to the PP. The axis of the frustum is 30
mm to the right of the eye. The eye is 45 mm
infront of the PP and 50 mm above the ground.
The nearest base corner is 10 mm behind the PP.
Draw the perspective projection of the frustum.

7
Draw the perspective view of a Pentagonal prism
of base side 20 mm and height 40 mm when it
rests on its base on the ground plane with one of
the rectangular faces parallel to and 20 mm
behind the picture plane. The station point is 45
mm infront of the PP and 60 mm above the GP.
The observer is 20 mm to the left of the axis.
Use top view and the end view to draw the
perspective by visual ray method.
8
A Hexagonal prism of base side 25 mm and axis
length 50 mm rests on the ground on one of its
rectangular faces with its axis inclined at 300 to
the picture plane. A corner of the base is
touching the PP. The station point is 60 mm
infront of the PP and lies in a central plane that
bisects the axis. The station point is 40 mm
above the ground plane. Draw the perspective
view of the prism.

Tutorial: XII
Free Hand Sketching
A
Draw the top view, front view and left side
views of the object shown in the diagram. All
the dimensions are in millimeters.

A
40
10
F
40
80
40
40
20
Left Side ViewFront View
Top View
B
Draw the top view, front view and left
side views of the object shown in the
diagram. All the dimensions are in
millimeters.
Left Side View
Top View
Front View

B
120
64
3X16
3030
90
100
Top View
C
Draw the top view, front view and
left side views of the object shown
in the diagram. All the dimensions
are in millimeters.
Top View

C
90
R15
φ1010
90
30
10 10
40
10
Left Side View
Front View
Top View
Draw the top view, front view and left
millimeters.
D
Draw the top view, front view and right
millimeters.
Right Side View
Front View
Top View

E
millimeters.
Right Side View Front View
Top View
F
millimeters.
Right Side View
Front View
Top View

G
left side views of the object
shown in the diagram. All the
dimensions are in millimeters.
Left side virwFront View
Top View
H
Draw the top view, front
view and right side views of
the object shown in the
diagram. All the dimensions
are in millimeters.
Top View

I
Draw the top view, front view
and right side views of the
object shown in the diagram.
All the dimensions are in
millimeters.
Top View
J
Draw the top view, front view and left side views of the object
shown in the diagram. All the dimensions are in millimeters.
Left Side View
Front View
Top View

K
Top View
right side views of the object
L
Top View
right side views of the object

Tutorial: XIII
SCALES
DIMENSIONS OF LARGE OBJECTS MUST BE REDUCED TO ACCOMMODATE ON
STANDARD SIZE DRAWING SHEET.THIS REDUCTION CREATES A SCALE OF THAT
REDUCTION RATIO, WHICH IS GENERALLY A FRACTION.. SUCH A SCALE IS
CALLED REDUCING SCALE
AND THAT RATIO IS CALLED REPRESENTATIVE FACTOR.
SIMILARLY IN CASE OF TINY OBJECTS DIMENSIONS MUST BE INCREASED FOR
ABOVE PURPOSE. HENCE THIS SCALE IS CALLED ENLARGING SCALE. HERE THE
RATIO CALLED REPRESENTATIVE FACTOR IS MORE THAN UNITY.
SCALES

SCALES
FOR FULL SIZE SCALE
R.F.=1 OR ( 1:1 )
MEANS DRAWING & OBJECT ARE OF SAME SIZE.
Other RFs are described as
1:10, 1:100,
1:1000, 1:1,00,000
= 1000 m
= 100 m
= 10 m
= 1 m
=
૚
૚૙
m
=
૚
૚૙૙
m
=
૚
૚૙૙૙
m
1 KILOMETRE
1 HECTOMETRE
1 DECAMETRE
1 METRE
1 DECIMETRE
1 CENTIMETRE
1 MILLIMETRE
BASIC UNITS.
1. PLAIN SCALES ( FOR DIMENSIONS UP TO SINGLE DECIMAL)
2. DIAGONAL SCALES ( FOR DIMENSIONS UP TO TWO DECIMALS)
3. VERNIER SCALES ( FOR DIMENSIONS UP TO TWO DECIMALS)
4. COMPARATIVE SCALES ( FOR COMPARING TWO DIFFERENT UNITS)
5. SCALE OF CORDS ( FOR MEASURING/CONSTRUCTING ANGLES)
6. ISOMETRIC SCALES
TYPES OF SCALES:
Not in
Syllabus
Not in
Syllabus
In Syllabus

1
Construct a diagonal scale using a scale of 10
centimeter to 6 meter to read meter, decimeter
and centimeter. Show a distance of 6.53 meter
on it.
2
Construct a diagonal scale of RF = 1/2 to
show millimeter and centimeter to
measure upto 35 centimeter. Show on the
scale a distance of 23.6 centimeter.

3
A distance of 30 cm measured on a map amounts to 450 meter. Draw a diagonal scale showing divisions
of 50 cm capable of measuring 300 meter. Mark on it 255.5 meter and 177.5 meter.
4
Draw a diagonal scale of RF. 1
/200000 to show kilometer and
decimals of a kilometer. Mark on the
scale a distance of 23. 7 kilometer.

5 A rectangular plot of land of area 45 hectare is represented on a map by a similar rectangle. of 5 sq. cm.
Calculate the RF. of the scale of the map. Also draw a diagonal scale to read upto a maximum of single km
from the map. The scale should be long enough to measure upto 500 km.
6
Construct a Vernier scale to read distances correct
to a decameter on a map in which the actual
distances are reduced in the ratio of 1: 40000. The
scale should be long enough to measure 6 kilometer.
Mark on the scale the lengths of 3.34 km and-0.57
km.

7
Construct a Vernier scale to give readings of
1/10th of a mm upto 100 mm and mark off on
it the following: (a) 81.3 mm (b) 20.9 mm.
Select a scale 1.5:1.
8 Construct a Vernier scale having a least count of 0.005 cm and long enough to measure 5 centimeter.
Show the following distances over the scale: (a) 3.15 cm (b) 1.95 cm. select a scale of 2:1.

CAT I Solutions
Section - ECE A
Staff Incharge – A.Kumaraswamy
1 Draw the projections of the following points on a common reference line with the distance between the end
projectors as 35 mm.
a. Point A is 30mm behind VP and on HP
b. Point Bis contained by both HP and VP
c. Point C is 25 mm below HP and 30mm behind VP
d. Point D is 25 mm below HP and 45 mm infront of VP
e. Point E is 30mm above HP and 25 mm infront of VP (10)

2a
A line LM 70 mm long has its end L, 10
mm above HP and 15 mm in front of VP.
Its top view and front view measures 60
mm and 40 mm respectively. Draw the
projections of the line and determine its
inclinations with HP and VP. Also locate
its traces. (20)
2b
The top view of a line AB, 80 mm long is
inclined at 45° to the xy line and measures
60 mm. The point A is located on HP and at
20 mm infront of VP. Draw the projections
of the line and find its inclinations. Also
locate its traces. (20)

3a A Pentagonal lamina of side 35 mm is resting upon its edge on HP, so that the surface is
inclined at 45° to HP. The line joining the midpoint of the resting edge to the opposite corner
is inclined at 30°to the VP. Draw the projections of the lamina
3b
A circle of 40 mm diameter rolls
inside the another circle of 200 mm
diameter for one revolution. Name
the curve and draw tangent and
Normal at any point on the curve.
(20)

CAT – II Solutions
Class : I sem ECE A Section
1a
A Hexagonal prism of side 30 mm and axis length 60 mm is resting
on HP on its base with two of its vertical faces perpendicular to VP.
It is cut by a plane inclined at 500 to HP and perpendicular to VP
and meets the axis of prism at a distance of 10mm from the top
end. Draw the development of the lateral surface of the prism.

2a
A Square pyramid of base side 30
mm and altitude 65 mm rests on the
HP on one of its base edges such
that the triangular face containing
the resting edge is perpendicular to
both HP and the VP. Draw its
projections.
Initial Position
411
2b
A cone of base diameter
50 mm and axis length 60
mm has one of its
generators on VP with the
solid axis parallel to and 40
mm above the HP. Draw its
projections.
Initial Position
412

3a
A hexagonal pyramid of base side 25mm and axis 55
mm rests on HP on its base with two edges parallel to
VP. It is cut by a vertical plane inclined at 30° to VP
and cutting the pyramid at a distance of 6 mm away
from the plan of the axis. Draw the top view, sectional
front view and true shape of the section..
Top view
True shape
3b
A cone of base diameter 50 mm and
altitude 60 mm rests on its base on HP. It
is cut by a section plane perpendicular to
VP and inclined to HP in such a way that
the true shape of the section is an
isosceles triangle of base 35 mm. Draw
the front view, sectional top view and true
shape of the section
Front view
Sectional top view
True shape

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