The document provides instructions for performing various geometric constructions using drawing instruments. It covers constructing lines, angles, triangles, quadrilaterals, circles, ellipses, parabolas, hyperbolas and their tangents. The methods include using a compass, set squares, concentric circles and the distance squared rule. Instructions are given step-by-step with diagrams to divide lines into ratios, bisect angles, construct perpendiculars, inscribe and circumscribe shapes, draw tangents and join two points with a curve. The document also introduces graphic language components, drawing instruments and their use in technical drawing and sketching.

1. Geometrical Constructions

2. 1. To divide a straight line into a given number of equal parts say
5.
1. Draw AC at any angle to AB.
2. Construct the required number of equal parts of
convenient length on AC like 1,2,3.
3. Join the last point 5 to B
4. Through 4, 3, 2, 1 draw lines parallel to 5B to
intersect AB at 4',3',2' and 1'.

3. 2. To divide a line in the ratio 1 : 3 : 4.
1. As the line is to be divided in the ratio 1:3:4
it has to be divided into 8 equal divisions.
2. Divide AC into 8 equal parts and obtain P
and Q to divide the line AB in the ratio 1:3:4.
3

4. 3. To draw a line through a given point, parallel to another line.
1. The line is to be drawn through given point C.
2. To draw a line through C parallel to AB, take D as
center on AB, and strike arc CE.
3. Shift the center to E, maintaining the same radius,
and strike arc DF.
4. Set a compass to a chord of arc CE, and lay off the
chord DF from D, thus locating point F.
5. A line drawn through F and C is parallel to AB.

5. 4. Drawing a line through a point and perpendicular to a line when the point is
not on the line
Method (a)
1. Draw from P any convenient inclined line as PD. Find
the midpoint C of line PD.
2. Draw arc with radius CP.
3. The line EP is the required perpendicular.
Method (b)
1. With P as center, strike an arc to intersect AB at C and D.
2. With C and D as centers, and with a radius slightly greater than
half CD, strike arcs to intersect at E.
3. The line PE is the required perpendicular.

6. 5. To bisect a given angle.
1. Draw a line AB and AC making the given angle.
2. With center A and any convenient radius R
draw an arc intersecting the sides at D and E.
3. With center D and E and radius larger than half
the chord length DE, draw arcs intersecting at F
4. Join AF, <BAF = <PAC.
A

7. 6. To transfer an angle
1. Angle BAC is to be transferred to the new position A' B'
2. Use any convenient radius R, and strike arcs from centers A and A‘
3. Strike equal arcs r, and draw side A' C'

8. 7. To draw an arc of given radius touching two straight lines at right
angles to each other.
1. Let r be the given radius and AB and AC the
given straight lines.
2. With A as center and radius equal to r draw arcs
cutting AB and AC at P and Q.
3. With P and Q as centers draw arcs to
4. meet at O. With 0 as center and radius equal to
r draw the required arc..

9. 8. To draw an arc of a given radius, touching two given straight lines making an
angle between them.
1. Let AB and CD be the two straight lines and r, the
radius.
2. Draw a line PQ parallel to AB at a distance r from
AB.
3. Similarly, draw a line RS parallel to CD.
4. Extend them to meet at O.
5. With 0 as center and radius equal to r draw the
arc to the two given lines.

10. 9. To draw a tangent to a circle
1. With 0 as center, draw the given circle. P is any point
on the circle at which tangent to be drawn
2. Join 0 with P and produce it to P' so that OP= PP'
3. With 0 and P' as centers and a length greater than OP
as radius, draw arcs intersecting each other at Q.
4. Draw a line through P and Q. This line is the required
tangent that will be perpendicular to OP at P.

11. 10. Drawing a triangle with sides given
1. Draw one side, as C, in desired position, and strike arc with radius
equal to side A.
2. Strike arc with radius equal to side B.
3. Draw side A and B from intersection of arcs, as shown in III.

12. 11. Drawing a right triangle with hypotenuse and one side given
1. Given sides S and R, with AB as a
diameter equal to S, draw a semicircle.
2. With A as a center and R as a radius,
draw an arc intersecting the semicircle
at C.
3. Draw AC and CB to complete the right
triangle

13. 12. Drawing an equilateral triangle
a. By a compass
b. By a 30º - 60º triangle

14. 13. To inscribe a square in a given circle.
1. With center 0, draw a circle of diameter D.
2. Through the center 0, draw two diameters,
say AC and BD at right angle to each other.
3. Join A-B, B-C, C- D, and D-A. ABCD is the
required square.

15. 14. To inscribe a square in a triangle.
1. Draw the given triangle ABC.
2. From C drop a perpendicular to cut the base AB at D.
3. From C draw CE parallel to AB and equal in length to CD.
4. Draw AE and where it cuts the line CB mark F.
5. From F draw FG parallel to AB.
6. From F draw FJ parallel to CD.
7. From G draw GH parallel to CD.
8. Join H to J.
9. Then HJFG is the required square.

16. 15a. To draw an external tangent to two circles of different diameters
1. Join the centers of circles a and b.
2. Bisect ab to obtain the center c of the
semicircle.
3. From the outside of the larger circle,
subtract the radius r of the smaller circle.
4. Draw the arc of radius ad. Draw normal Na.
5. Normal Nb is drawn parallel to normal Na.
6. Draw the tangent.

17. 15b. To draw an internal tangent to two circles of different diameters
1. Join the centers of circles a and b.
2. Bisect ab to obtain the center c of the
semicircle.
3. From the outside of the larger circle, add
the radius r of the smaller circle.
4. Draw the arc of radius ad. Draw normal
Na.
5. Normal Nb is drawn parallel to normal Na.
6. Draw the tangent.

18. 16a. To draw a tangential arc to two circles of different diameters
with outside radius
1. Two circles of radii a and b are tangential
to arc of radius R.
2. From the center of circle radius a,
describe an arc of R + a.
3. From the center of circle radius b,
describe an arc of R + b.
4. At the intersection of the two arcs, draw
arc radius R.

19. 16b. To draw a tangential arc to two circles of different diameters with
inside radius
1. Two circles of radii a and b are
tangential to arc of radius R.
2. From the center of circle radius a,
describe an arc of R - a.
3. From the center of circle radius b,
describe an arc of R - b.
4. At the intersection of the two arcs,
draw arc radius R.

20. 17. To construct a pentagon, given the length of side.
1. Draw a line AB equal to the given length of side.
2. Bisect AB at P.
3. Draw a line BQ equal to AB in length and perpendicular to AB.
4. With center P and radius PQ, draw an arc intersecting AB produced
at R. AR is equal to the diagonal length of the pentagon.
5. With centers A and B and radii AR and AB respectively draw arcs
intersecting at C.
6. With centers A and B and radius AR draw arcs intersecting at D.
7. With centers A and B and radii AB and AR respectively draw arcs
intersecting at E.
ABCDE is the required pentagon.
D
E

21. 18 To construct a hexagon, given the length of the side using set
square
1. Draw a line AB equal to the side of the hexagon.
2. Using 30° - 60° set-square draw lines A1, A2, and
B1, B2.
3. Through 0, the point of intersection between the
lines A2 at D and B2 at E.
4. Join D,E
5. ABCDEF is the required hexagon.

22. 19. To construct a hexagon, given the length of the side using compass
1. Draw a line AB equal to the of side of the hexagon.
2. With center A and B and radius AB, draw arcs
intersecting at 0, the center of the
hexagon.
3. With center 0 and B and radius OB (=AB) draw arcs
intersecting at C.
4. Obtain points D, E and F in a similar manner.

23. 20. To inscribe a hexagon in a given circle.
1. With center 0 and radius R draw the given circle.
2. Draw any diameter AD to the circle.
3. With centers A and D and radius equal to the radius of
the circle, draw arcs intersecting the circles at B, F, C and
E respectively.
4. ABC D E F is the required hexagon.

24. 21. To circumscribe a hexagon on a given circle of radius R.
1. With center 0 and radius R draw the given circle.
2. Using 30-60 set square, circumscribe the hexagon as
shown.

25. 22. To construct an inscribed regular polygon having any number
of sides with the given diameter of the circle.
1. Draw a circle with the given diameter.
2. Divide its diameter into the required number of equal parts
(seven in this example).
3. Use the inclined line method to divide the line.
4. With a radius equal to the diameter and with centers at the
diameter ends (Points A and B), draw arcs intersecting at
Point P.
5. Draw a line from Point P through the second division point
of the diameter (Line AB) until it intersects with the circle at
Point C. The
6. second point will always be the point used for this
construction. Chord AC is one side of the polygon.
7. Lay off the length of the first side around the circle using
dividers. This will complete the regular polygon with the
required number of sides.

26. 23. To draw any regular polygon with a given length of a side
• Draw a line AC with the given length and extend AC to
B, making CB equal to AC.
• With C as center and AC as a radius, draw a semicircle.
• Divide the semicircle into 8 equal parts from A to B, and
draw radii from C to the points of intersection on the
semicircle.
• The radius C6 is always the second side of the polygon.
• Draw a circle through points A, C and 6..
• The circle drawn is the circumscribed circle of the
polygon.
• To draw the remaining sides, extend the radii from the
semicircle and connect the points where they intersect
the circumscribed circle.

27. 24. To construct a regular figure of given side length and of N sides on a straight line.
1. Draw the given straight line AB.
2. At B erect a perpendicular BC equal in length to AB.
3. Join AC and where it cuts the perpendicular bisector of AB, number the point
4.
4. Complete the square ABCD of which AC is the diagonal.
5. With radius AB and center B describe arc AC as shown.
6. Where this arc cuts the vertical center line number the point 6.
7. This is the center of a circle inside which a hexagon of side AB can
now be drawn.
1. Bisect the distance 4-6 on the vertical center line.
2. Mark this bisection 5. This is the center in which a regular pentagon of side AB
can now be drawn.
3. On the vertical center line step off from point 6 a distance equal in length to
the distance 5-6. This is the center of a circle in which a regular heptagon of
side AB can now be drawn.
4. If further distances 5-6 are now stepped off along the vertical center line and
are numbered consecutively, each will be the center of a circle in which a
regular polygon can be inscribed with side of length AB and with a number of
sides denoted by the number against the center.

28. Complex Geometry - Planes

29. 1a. Drawing an ellipse by the four-center method
1. Given major and minor axes, AB and CD, draw line AD connecting the end points as shown.
2. Mark off DE equal to the difference between the axes AO - DO.
3. Draw perpendicular bisector to AE, and extend it to intersect the major axis at K and the minor axis extended at H.
4. Mark off OM equal to OK, and OL equal to OH. The points H, K, L and M are the centers of the required arcs.
5. Using the centers, draw arcs as shown. The four circular arcs thus drawn meet in common points of tangency P at
the ends of their radii in their lines of centers.

30. 1b. Drawing an ellipse by oblong method
A
1. Draw the major and minor axes AB and CD and
locate the center O.
2. Draw the rectangle KLMN passing through A,D,B,C.
3. Divide AO and AN into same number of equal parts,
say 4.
4. Join C with the points 1',2',3' .
5. Join D with the points 1,2,3 and extend till they meet
the lines C1, C2, C3 respectively at P1, P2 and P3
6. Repeat steps 3 to 5 to obtain the points in the
remaining three quadrants.
7. Join the points by a smooth curve forming the
required ellipse.

31. 1c. Drawing an ellipse by concentric circle method
1. Draw the major and minor axes AB and CD and locate
the center O.
2. With center 0 and major axis and minor axes as
diameters, draw two concentric circles.
3. Divide both the circles into equal number of parts, say
12 and draw the radial lines.
4. Considering the radial line 0-1'-1, draw a horizontal line
from 1' to meet the vertical line from 1 at P1
5. Repeat the steps 4 and obtain other points P2, P3, etc.
6. Join the points by a smooth curve forming the required
ellipse.
0

32. 3. Drawing a parabola by the distance squared method
1. This method is based on the fact that the parabola may be
described by the equation y=ax2.
2. Given the rise AD, and span AB of the parabola, bisect AB
at O and divide AO into a number of equal parts.
3. Divide AD into a number of equal parts amounting to the
square of the number of divisions of AO.
4. From line AB, each point on the parabola is offset by a
number of units equal to the square of the number of
units from point O.
5. For example, point 3 projects 9 units.
6. This method is generally used to draw parabolic arcs.

33. 2. Drawing a parabola by the parallels to directrix method
1. The parallel directrix method is based on the fact that for each
point on a parabola, the distance from the focus is equal to the
distance from the directrix.
2. Given a focus F and a directrix AB, draw line DE parallel to the
directrix at any distance CZ from it.
3. With center at F and radius CZ, strike arcs to intersect the line DE
in the points Q and R, which are points on the parabola.
4. Determine as many additional points as are necessary to draw
the parabola accurately, by drawing additional lines parallel to
the directrix and proceeding in the same manner.

34. 4. Joining two points by a parabolic curve.
1. Let X and Y be the given points. Assume any point O, and draw tangents XO and YO.
2. Divide XO and YO into the same number of equal parts, number the division points as shown, and
connect the corresponding points.
3. These lines are tangents of the required parabola, and form its envelope. Use to sketch a smooth
curve.

35. 5. Drawing a hyperbola by the geometric method.
1. Select any point X on the transverse axis.
2. With centers at F and F', and BX as radius, strike the
arcs DE.
3. With same centers and AX as radius, strike arcs to
intersect the arcs first drawn in the points Q, R, S and
T, which are points on the required hyperbola.
4. By selecting a different location for the point X, find
as many additional points as necessary to draw the
curve accurately.

37. Introduction to drawing instruments ..
• Methods of employing them for technical drawing and sketching.
Graphic language and its components..
• Line types: meaning and application
• Architectural Lettering and dimensioning techniques
• Architectural annotations and conventions including representation of various building materials
and building components
• Graphic scales and their application
Plane and Solid geometry..
• Introduction to graphical construction of various plane geometrical shapes
• Introduction to various projection systems used in Architectural drawing; such as Orthographic,
Isometric and Axonometric projections to draw and represent various three dimensional
geometrical objects/forms including Section/s.
Scale Drawing..
• Scale drawing (plan/s section/s and elevation/s) of a simple building of sufficient size to
demonstrate use of various metric scales, conventions and standard annotations.
• Principles of free hand sketching such as proportions, light and shade; with primary thrust on
sketching of building elements and built environment.
Sketching..
Course Contents