development

1. 1
MEL 110
Development of surfaces

2. 2
Prism – Made up of same number of rectangles as sides of the base
One side: Height of the prism
Other side: Side of the base
Cylinder – Rectangle
One side: Circumference of the base
Other side: Height of the cylinder
Pyramid – Number of triangles in contact
The base may be included
if present
Development is a graphical method of obtaining the area of the surfaces of a solid.
When a solid is opened out and its complete surface is laid on a plane, the surface of
the solid is said to be developed. The figure thus obtained is called a development of
the surfaces of the solid or simply development. Development of the solid, when folded
or rolled, gives the solid.
h
d
d
Examples
T. L.

3. 3
Methods used to develop surfaces
1. Parallel-line development: Used for prisms, cylinders etc. in which
parallel lines are drawn along the surface and transferred to the
development.
2. Radial-line development: Used for pyramids, cones etc. in which the
true length of the slant edge or generator is used as radius.
3. Triangulation development: Complex shapes are divided into a
number of triangles and transferred into the development (usually
used for transition pieces).
4. Approximate method: Surface is divided into parts and developed.
Used for surfaces such as spheres, paraboloids, ellipsoids etc.
Note:- The surface is preferably cut at the location where the edge will be
smallest such that welding or other joining procedures will be minimal.

4. 4
Parallel line development: This method is employed to develop the surfaces of prisms
and cylinders. Two parallel lines (called stretch-out lines) are drawn from the two
ends of the solids and the lateral faces are located between these lines.

5. 5
D
H
D
S
S
H
L

 = R
L
+
3600
R=Base circle radius.
L=Slant height.
L= Slant edge.
S = Edge of base
L
S
S
H= Height S = Edge of base
H= Height D= base diameter
Development of lateral surfaces of different solids.
(Lateral surface is the surface excluding top & base)
Prisms: No.of Rectangles
Cylinder: A Rectangle
Cone: (Sector of circle) Pyramids: (No.of triangles)
Tetrahedron: Four Equilateral Triangles
All sides
equal in length
Cube: Six Squares.
Parallel-line
development
Radial-line
development

6. 6
L L

 = R
L
+
3600
R= Base circle radius of cone
L= Slant height of cone
L1 = Slant height of cut part.
Base side
Top side
L1 L1
L= Slant edge of pyramid
L1 = Slant edge of cut part.
DEVELOPMENT OF
FRUSTUM OF CONE
DEVELOPMENT OF
FRUSTUM OF SQUARE PYRAMID
FRUSTUMS
FRUSTUMS

7. 7
Project, horizontally, the points of intersection of the
cutting plane with the edges.
Mark distances 3M, 3N
2, b
4, d
1
2
3
4
A
B
C
1
D
Cube cut by section
plane

8. 8
2
Draw the development of the lower portion of the cone surface cut by a plane. Cone base diameter is 40 mm and
height is 50 mm. The cutting plane intersects the cone axis at an angle of 45o
and 20 mm below the vertex
a
b
c
d e
f
a
b c d e f g g
4
2
3
1
o
4
3
1
a
b
c
d
e
f
g
o
T
F
l
Length of arc =
circumference of
base of cone
2
b
2
True lengths b2, 2o
obtained by
auxiliary view
method
o
• Divide the cone in the top view and project the corresponding
generator lines in the front view
• Develop the complete surface of the cone by drawing an arc with
radius = length of side generator of cone and length of arc =
circumference of cone base
• Draw the corresponding generator lines
• Obtain true lengths of o1, o2 etc. by auxiliary view, rotation method
OR by projecting onto one of the side generators (which are in true
length)
• Mark the distances (true lengths) o1, o2…etc. in the development and
join them to get the development of the lower portion of the cone
a
o
l
 = R
l
+
3600
Radius of cone = R
2’
3’
4’
True length of (o2, o3) = (o2’, o3’) etc.

9. 9
If R = 2r then θ = 180°, i.e., if the slant height of a cone is equal to its
diameter of base then its development is a semicircle of radius equal
to the slant height.

10. 10
Develop the surface of the symmetrical half of an oblique pyramid with a horizontal
regular hexagonal base (side 20 mm and vertex 30 mm above one corner of the base)
o
a b c d
a
b c
o, d
c b
c
b
True lengths
o
c
b
a
d
T
F
Obtain true lengths of the edges ob and oc by rotation or auxiliary view
method
Edge oa is seen in true length in the Front View
ab = bc = cd = side of hexagonal base = 20 mm
od and dc can be constructed as they are
perpendicular to each other
The lengths of bc, and ob are known and therefore
these distances can be marked with the compass
After drawing triangles odc and ocb, triangle oba
can be completed
a
d

11. 11
a b c
Develop the surface of the cylinder which is cut as shown
a
b,l c,k d,j
e,i f,h
g
d
e
f
g
• Divide the base of the cylinder in the top and front views into
the a certain number of equal parts (12 here)
• Develop the surface of the cylinder (rectangle with length x
diameter and height = height of cylinder) and divide it into the
same number of equal parts
• The projector lines from the top view intersect the cut portion
of the cylinder at a, b, c…..f.
• Project these points onto the developed surface
15o
45o
x50
100
T
F
h i h h
a
a g
b
c
d e
f
h
i
j
k
l
h
i
j
k l
50
30o

12. a’, e’ b’, f’
d’, h’ c’, g’
a, i, d b, k, c
e, j, h f, l, g
i’, j’
k’, l’
b
i’, j’ b’, f’
d’, h’ k’, l’
j f
i
d
h
k
l
Oblique square prism
i’ b’
d’ k’
f’
l’
j’
h’
i’
d’

13. 13
Oblique prism
a b
c
d
e
f
a b
f
h i
g
h i
g
a
h
b
i
f
g
f
g
Parallel to each other