UNIT 4:DEVELOPMENT OF

SURFACES
18ME2ICEED

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 1

Development of Surfaces. A development is the unfold / unrolled flat / plane figure of a
3-D object. In other words, the development of a solid is the shape of a plain sheet that
by proper folding could be converted into the shape of the concerned solid as shown in
figure.

It is also called a pattern where the plane may show the true size of each area of the
object. When the pattern is cut, it can be rolled or folded back into the original object

If he face of a olid i laid o on a plain face he hape h ob ained i called

he de elopmen of ha olid
Surface development is the process of mapping a given 3D surface into a 2D shape

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 2

 Applications: Knowledge of development is very essential in
sheet metal industries

Storage tanks :Oil refineries Piping industry

Boilers Ship building

Other applications :Chimneys,Hopper,,Aircraft Industry
4-May-20 3
DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19
 PRINCIPLES OF SURFACE DEVELOPMENT
Every line on the development should show the true length of the corresponding
line on the surface which is developed.
METHODS OF DEVELOPMENT
1.PARALLEL LINE DEVELOPMENT 2.RADIAL LINE DEVELOPMENT
Applies for developing Prisms and Employed for Pyramids and single
single curved surfaces like curved surfaces like Cones in
Cylinders, in which all the which the apex is taken as centre
edges/generation of lateral and the slant edge or generator as
surfaces are parallel in each other. radius of its development.

It is used for developing Prisms

and single curved surfaces like
Cylinders, in which all the
edges/generation of lateral
surfaces Square
are parallel
Prismin each other. Square Pyramid

Cylinder Cone
4-May-20 4
DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19
 METHODS OF DEVELOPMENT

3.TRIANGULATION DEVELOPMENT 4.APPROXIMATE DEVELOPMENT

Used in developing Transition pieces Employed for Doubly curved surfaces
Which are employed to connect two like Sphere
different forms such as square to Theoretically not possible to develop,
round etc. hence approximate
Development is done by assuming the
surface to be made from a series of
triangular surfaces laid side by side.i.e
Triangulation

Zones

Square to round: development of one half

DEPARTMENT OF MECHANICAL
4-May-20 5
ENGINEERING, BMSCE, BANGALORE-19
 1.PARALLEL LINE DEVELOPMENT

Square Prism(Folding edge) Cylinder( No Folding edge)

2.RADIAL LINE DEVELOPMENT

Square Pyramid (Folding edge) Cone (No Folding edge)

4-May-20 6
DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19
 SECTION OF SOLIDS
In some cases ,objects are cut based on designer requirement. These cutting planes
are called section planes

Section views are very important to reveal interior features of parts which are not
otherwise not visible. Section of solids are based on the use of Imaginary cutting
plane .

Trace of a section plane is a line in which the section plane meets the reference
plane. The projection of the section on the reference plane to which the section
plane is perpendicular will be straight line coinciding with the trace of the section
plane on it.
DEPARTMENT OF MECHANICAL
4-May-20 7
ENGINEERING, BMSCE, BANGALORE-19
• Vertical trace (V.T.) – V.T. of a section plane is a line in
which the section plane meets the V.P.
• Horizontal trace (H.T) – H.T. of a section plane is a line in
which the section plane meets the H.P
• Section plane intersection points are denoted with
numbers
• Types of Section plane :These cutting planes can be
1. Vertical Section Plane cutting plane parallel to vertical
plane(HT)

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 8

2. Horizontal Section Plane cutting plane
parallel to horizontal plane(VT)

Frustrum

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 9

3.Auxiliary Inclined Plane: A plane inclined to
horizontal plane and perpendicular to vertical
plane(VT)

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 10

 4.Auxiliary Vertical Plane: A plane inclined to vertical plane
and perpendicular to horizontal plane(HT)

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 11

Frustum: When the cutting plane is parallel to the base
plane of a prism, pyramid or cone, The cut portion is called the
frustum of the solid. The portion between the observer and the
cutting plane is assumed to be removed and the portion which is
left is called the Frustum of a solid

Frustum of cone & pyramids.

( top & base parallel to each other)

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 12

 Truncated Solids:
• When the section plane is inclined to the base of a solid, the cut portion
is called truncated solid. The portion between the observer and the
cutting plane is assumed to be removed and the portion which is left is
called the truncated solid.

Sections of solids( top & base not parallel)

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 13

 SECTION POINTS
Example 1.A square prism of base side 30mm and axis length 60mm is resting on HP on its base with all the
vertical faces being equally inclined to VP, It is cut by an inclined plane 600 to HP and perpendicular to VP
and is passing through a point on the axis at a distance 45mm from the base. Draw the section plane and
identify section points

Fig 1B

Fig 1A
Fig 2
4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 14
Example 2.A pentagonal prism of base sides 20mm and height 40mm is resting with its base on HP and base
edge parallel to the VP. The prism is cut as shown in the following front view. Draw the section plane and
identify section points

Fig 1B

Fig 2
Note:When two section planes intersect,
Two section points ,one at front and another at back
Will be formed.In this problem section points 2 and 5
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 15

1) A triangular prism with one of its rectangular faces parallel to VP and nearer to it is cut as shown in fig.
Draw the developments of the retained portions of the prism which are shown in the dark line
SOLUTION:

(8')
11 11 (6')
a' 2' b' 4' c' A 2 B 4 C 6 8 A
Fig 1B 15

1' 1
3' (7') 5' 1 3 5 7
60

DEVELOPMENT

VP
X Y
HP p' q' r' P Q R P

120

(p) a 1 8 7 6 5 c (r)
Fig 2

2 4
40

3
Fig 1A b (q)

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 16

 2. A square prism of base 35mm and height 60mm is resting on its base, such that one of the base edges
is inclined at 20° to VP. It is cut by a section plane, which is perpendicular to VP, inclined at 60° to HP and
passes through the axis at a height of 20mm. Draw the development of the base portion of the lateral
surface.

Fig 1B

3' 3
60

(4') 4

2' 2
20

1' 1 1

D A
20°

a' b' (d') c' A B C

d
35
a 140
35

c
Fig 2
b

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 17

 3) A square prism of base side 30mm and axis length 60mm is resting on HP on its base with all the vertical
faces being equally inclined to VP, It is cut by an inclined plane 600 to HP and perpendicular to VP and is
passing through a point on the axis at a distance 45mm from the base. Draw the development of the lower
portion of the prism

Fig 2
Fig 1B

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 18

4) A square prism of base side 40mm and axis length 65mm resting on HP on its base with all the vertical
face being equally inclined to VP. It is cut by an inclined plane 600 to HP and perpendicular to VP and is
passing through a point on the axis at a distance 15mm from the top face. Draw the development of the
lower portion of the prism

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 19

5) A square prism of 30mm side of the base and height 50mm is resting with its base on HP such that one of
its vertical faces is inclined at 400 to VP. It is cut as shown in the following front view fig. Draw the
development of the lateral surface of the prism

Fig 1B

Fig 2
Note:Development should be started about cc1 and not
AA1,else It will not be a single piece development

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 20

6 A cube of side 40 mm is resting on HP with its base on HP such that one of its vertical
aces is inclined at 30 deg to VP. It is cut by a section plane perpendicular to VP ,inclined
to HP at an angle 45 deg and passes through midpoint of the axis. Draw the
development of the lower lateral surface of cube

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 21

7.A square prism of 35mm side of the base and height 50mm is resting with
its base on HP such that one of its vertical faces are equally inclined to VP. It
is cut as shown in the following front view fig. Draw the development of the
lateral surface of the prism
Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 22

8) A rectangular prism of base 40mmX25mm and height 65mm rests on HP on its base with the longer base
side inclined at 300 to VP. It is cut by a plane inclined at 400 to HP, perpendicular to VP cuts the axis at its mid
height. Draw the development of the remaining portion of the prism

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 23

9) A rectangular prism base 30mm X 20mm and height 60mm rests on HP on its base with the longer base
side inclined at 400 to VP. It is cut by a plane inclined at 450 to HP, perpendicular to VP and bisects the axis.
Draw the development of the lateral surface of the prism

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 24

10) A rectangular prism of base size 25mm X 40mm and axis length 65mm is resting on HP on its base with
the longer side of base inclined at 300 to VP. It is cut by a plane inclined at 400 to HP and perpendicular to VP
and passes through the extreme left corner of base. Draw the development of the lateral surface of the
remaining portion of the prism
Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 25

11) Draw the developments of the truncated portion of the lateral faces of a pentagonal prism of 20mm sides
of base and 50mm height standing vertically with one of its rectangular faces parallel to VP and nearer to it so
as to produce a one piece development. The inclined face of the truncated prism is 300 to its axis and passes
through the right extreme corner of the top face of the prism

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 26

12) A rectangular pentagonal prism of height 60mm and base edge 30mm rests with its base on HP. The
vertical face closest to VP is 300 to it. Draw the development of the truncated prism with its truncated surface
inclined at 600 to its axis and bisecting it.

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 27

13) A pentagonal prism of 30mm side of base and height 50mm lies with its base on HP such that one of the
rectangular faces is inclined at 400 to VP. It is cut to the shape of a truncated pyramid with the truncated
surface inclined at 300 to the axis so as to pass through a point on it 30mm above the base. Develop the
truncated portion of the prism so as to produce a one piece development.

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 28

14) A pentagonal prism of base sides 30mm and axis length 60mm rests with its base on HP and an edge of
the base inclined at 450 to VP. It is cut by a plane perpendicular to VP, inclined at 400 to HP and passing
through a point on the axis, at a distance of 30mm from the base, Develop the remaining surfaces of the
truncated prism

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 29

15) A pentagonal prism of base sides 20mm and height 40mm is resting with its base on HP and base edge
parallel to the VP. The prism is cut as shown in the following front view. Draw the development of the lateral
surface of the prism

Fig 1B

Fig 2

Fig 1A

Note:When two section planes intersect, two section

points ,one at front and another at back
will be formed.In this problem section points 2 and 5

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 30

 16. A pentagonal prism 30mm side of base and axis 60mm stands vertically with one
of its rectangular faces parallel & nearer to VP. Two semicircular holes of diameter
40mm is drilled through lateral faces such that the top face and base of the prism
contain the flat portions of the semicircular hole and their axis intersect the axis of
the prism at right angles. Draw the development of the lateral faces of the prism
showing the true shapes of the holes on it.

0
R2
Fig 1B

R
20
60

4'(12') 12
3' (13') 5' (11') 11 13
(14') (10') 4 5 10 14
6'(9') 3 15
(15')2'
2 6 9
a' 1'(16')(e') b' (d') 7' c' A 1 B 7 C 8 D E 16 A
5 (8')

R 20
e 5,28 13,82
d 13,82
8,59
22,49

22,49
17,64
13,82

30
5,2 a 150
c
8
8,5
9 Fig 2
17,6 b
4
Fig 1A
30

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 31

 17. A vertical hexagonal prism of 30 mm side of base and axis 65 mm long has one
of its rectangular faces parallel to VP and nearer to it. A circular hole of 40 mm
diameter is drilled through the prism completely such that the axis of the hole
bisects the axis of the prism at right angles and is perpendicular to VP. Draw the
development of the prism showing the shape of the hole on it.

Fig 1B R
20
R
20

10' 10

1' 1 1
65

9' 9 9
2' 8' 2 8 2
3' 7' 3 7
4' 6' 4 6
5' 5

a' b' (f') (o') c' (e') d' A B C D E F A

5

R 20

R 20
f e
O

10 5
40

30 15
180

a d

Fig 2
30

c
10
5

b
5
Fig 1A 15

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 32

18) A hexagonal prism of base side 20mm and height 50mm is resting on HP on its base, such that one of its
base edge is parallel to VP, The prism is cut in this position as shown in the following front view, Draw the
development of the lateral surface of the prism

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 33

19) A hexagonal prism of base side 25mm and height 55mm is resting on HP on its base, such that one of its
base edges is parallel to VP, the prism is cut in this position as shown in the following front view, Draw the
development of the lateral surface of the prism

Fig 1B

Fig 2

Note:Development should be started about DD1 and not

Fig 1A AA1,else It will not be a single piece development

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 34

20) The inside of hopper of a flour mill is to be lined with thin sheet. The top and bottom of the hopper are
regular pentagons with each side equal to 30mm and 22.5mm Respectively. The height of the hopper is
30mm. Draw the shape of the sheet to which it is to be cut so as to fit into the hopper

Fig 1B

Fig 2

Fig 1A
4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 35
21) A square pyramid of side of base45mm, altitude 70mm is resting with its base on HP with two sides of the
base parallel to VP. The pyramid is cut by a section plane which is perpendicular to the VP and inclined at 400
to the HP. The cutting plane bisects the axis of the pyramid. Obtain the development of the lateral surfaces the
truncated pyramid
Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 36

22) A square pyramid base 40mm side and axis 65mm long has its base on HP and all the edges of the base
are equally inclined to VP. It is cut to with an inclined section plane so as the truncated surface at 450 to its
axis bisecting it. Draw the development of the truncated pyramid
Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 37

23. A square pyramid of side of base45mm, altitude 70mm is resting with its base
on HP with two sides of the base parallel to VP. The pyramid is cut by a section
plane which is perpendicular to the VP and inclined at 400 to the HP. The cutting
plane bisects the axis of the pyramid. Obtain the development of the lateral
surfaces the truncated pyramid
Fig 1B o'

30,2
45

8
52,6
D

5
C

76,8
(3') 2'

40°

9
DEVELOPMENT
70

A
(4') 1' 4 B
35

1
3
(d') a' (o 1') (c') b' b1 '
52

30,28
,65 2
d c

4 A
O
3
Fig 2 76,89
1

(o1 ) b1
o
2
1

a b
q 45 Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 38

24) A frustum of a square pyramid has its base 40mm sides, top 16mm sides and height 60mm, its axis is
vertical and a side of its base is parallel to VP. Draw the projections of the frustum and show the development
of the lateral surface of it.

Fig 1B

Fig 2
Fig 1A
4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 39
25) A square pyramid of 25mm base edge and 50mm height rests with its base on HP with all of its base
edges equally inclined to VP. It is cut by a plane perpendicular to VP and inclined to HP at 600, passing
through the extreme right corner of base. Draw the development of the lateral surface of the pyramid.

Fig 1B

Fig 2
Note:Development should be started about Oc
and not OA,else It will not be a single piece
development
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 40

 26. A rectangular pyramid, side of base 25mm X 40mm and height 50mm has one
of the sides of the base is inclined at 300 to the VP. Draw the development of the
lateral surface of the cut pyramid, whose front view is shown in fig

Fig 1B

o'

15,2
19,9
2
8
4 D
27,6
4' C

55,2
(5') (5T')

60°
40

8
1' 2' 3'
50

(6') A
DEVELOPMENT
25

6 B

5
c' c ' 1 3 2

15,22
1

b' (o 1') (d')

25
a' 4
30°

19,98
d 1
O A
6 27,64
5
a o c 1 55,28
1 (o1 ) 4 c
2 Fig 2
25

3
40
b
Note:When two section planes intersect,
Fig 1A Two section points ,one at front and another at
back Will be formed(3,6)
4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 41
 27) A frustum of a pentagonal pyramid, smaller base sides 16mm and bigger top face sides 32mm and height
40mm, is resting on the HP on its smaller base, with one of its base sides parallel to the VP. Draw the
projections of the frustum and develop the lateral surface it.

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 42

28. The inside of hopper of a flour mill is to be lined with thin sheet. The top and
bottom of the hopper are regular pentagons with each side equal to 30mm and
22.5mm Respectively. The height of the hopper is 30mm. Draw the shape of the
sheet to which it is to be cut so as to fit into the hopper
(e') a' (o') b' (d') c'

Fig 1B 30

p' (t') (o ') q' (s') r'

1 E

P D
DEVELOPMENT

122,68 T
92,01

S C

22,5
B

d Q Fig 2

30
e
s
t
92,01 P A
22,5
30

o (o ) r c
1 122,68
p
q
a

Fig 1A b

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 43

 29) A regular pentagonal pyramid of side base 35mm and altitude 65mm has its base on HP with a side of
base perpendicular to VP. The pyramid is cut by a section plane which is perpendicular to the VP and inclined
at 300 to HP. The cutting plane meets the axis of the pyramid at a point 30mm below the vertex. Obtain the
development of the remaining part of the pyramid.

Fig 1B

Fig 2

Fig 1A
4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 44
30) A pentagonal pyramid, 30mm sides, with a side of base perpendicular to VP. Draw the development of the
lateral surfaces of the retained portion of the pyramid shown by the dark lines in the following figure.

Fig 1B

Fig 2
Note:When two section planes intersect,
Two section points ,one at front and another at
back Will be formed(2,6)

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 45

31. A Pentagonal pyramid of 30 mm edge of base and 55 mm high vertically rests with one
of its base edges parallel to VP and nearer to it. It is cut by two section planes, both being
perpendicular to VP. One of the section planes is horizontal and cuts the portion of the
pyramid on the left of the axis at a height of 20 mm above the base of the pyramid. The
other section plane inclined at 45 degrees to HP cuts the portion of the pyramid to the right
of the axis passing through a point on it 20 mm above the base and leans upwards. Draw
the development of the lower portion of the pyramid.

(o ')

Note:When
30,3

38,5
26,3 A

60,6
2 ,63

8
6
Fig 1B 60 two section

3
3'
1
(4')
planes
55

45°

1' (6') (5')

38,58
2' 6 E intersect,
30,32
20

o 5 Two section
4
e' (o ')b' (d') c' c ' points ,one at
38,58
a' 1
1

26
,36
3
d D
front and
e
5 4 O 1 2
another at
6

o
back
c
a
(o1 ) 1

C Will be
1 3 c
A B formed(2,5)
Fig 1A 30
2
30 Fig 2
b

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 46

32) A pentagonal pyramid of 30mm edges of base and 50mm height rests vertically with one of its base edges
parallel to VP and nearer to it. It is cut as shown in following figure. Draw the development of the lateral
surfaces of the upper portion of the pyramid.

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 47

33) A hexagonal pyramid, base sides 25mm and height 60mm, is resting with its base on HP and an edge of
base inclined at 400 to VP. It is cut to the shape of a truncated pyramid with the truncated surface indicated in
the front view at a point on the axis 20mm from the apex and inclined at 400 to XY. Draw the projections and
show the development of the lateral surface of the remaining portion of the pyramid.

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 48

34) A hexagonal pyramid of sides 35mm and altitude 65mm is resting on HP on its base with two of the base
sides perpendicular to VP. The pyramid is cut by a plane inclined at 300 to HP and perpendicular to VP and is
intersecting the axis at 30mm above the base. Draw the development of the remaining portion of the pyramid.

Fig 1B

Fig 2

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 49

35) A hexagonal pyramid 25mm side of base and axis 65mm long is resting on its base on HP with one of the
edges of the base parallel to VP. It is cut by a vertical section plane at a distance of 8mm from the axis
towards right side. Develop the lateral surface of the left part of the pyramid

Fig 1B

Fig 2
Note:Development should be started about OD and not
OA,preferably

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 50

36) A hexagonal pyramid of 30mm base sides with a side of base parallel to VP. Draw the development of the
lateral surfaces of the retained portions of the pyramid cut by two perpendicular planes shown by dark lines in
the Fig.

Fig 1B

Fig 2
Note:When two section planes intersect,
Two section points ,one at front and another at
back Will be formed(1,5)
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 51

37) A hexagonal pyramid of 30mm sides of base with a side of base parallel to VP. Draw the development of
the lateral surfaces of the retained portion of the pyramid which is shown by dark lines in the following figure.

Fig 1B

Fig 2

Note:When two section planes intersect,

Two section points ,one at front and another at
back Will be formed(1,5)
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 52

38) A vertical cylinder of base diameter 45mm and axis length 60mm is cut by a plane perpendicular to VP
and inclined at 500 to HP, is passing through the centre point of the top face. Draw the development of the
lateral surface of the cylinder.

Fig 1B

Fig 2B

2A
D=3.14x45=141.36mm

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 53

 39. Figure 4 shows the front view of a model of a steel chimney of diameter 60mm
made from a flat thin sheet metal fitted over an inclined plane roof. Develop the
portion of the chimney.

Fig 1B 5 5

(m') (l') (k') (j') (h')

a' b' c' d' e' f' g' A B C D 5 E F G H J 11 K L M A

DEVELOPMENT

(11')
45
(12') 5' 5 11
4' 6' (10') 6 10 12
4
7 9
3' 120° 8
7' 8' 3 13
1' 2' (13') (9') 2 14
(14')
18 1 1
188.4
k 11
l j
P= X D=188.4
m h
Fig 2B
a g

2A
b f
D=3.14x54=161.67 mm
c e
d 5 5
Fig 1A O 60

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 54

40) A vertical cylinder of base diameter 50mm and axis length 60mm is cut by a two planes which are
perpendicular to VP and inclined at 450 to HP and passing through either side the centre point of the top face.
Draw the development of the lateral surface of the cylinder

Fig 1B

Fig 2B

2A
D=3.14x50=157mm
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 55

41) Develop the lateral surface of the cylinder of 40mm diameter and height 60mm which is cut in the
following way.

Fig 1B

Fig 2B

2A
D=3.14x40=125.6mm
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 56

42) Develop the lateral surface of the cylinder of 40mm diameter and height 60mm which is cut in the
following way.

Fig 1B

Fig 2B

2A
D=3.14x40=125.6mm
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 57

43) Draw the development of the lateral surface of a truncated vertical cylinder, 40mm diameter of base and
height 50mm, the truncated flat surface of the cylinder bisects the axis at 600 to it.

Fig 1B

Fig 2B

2A
D=3.14x40=125.6mm
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 58

44) Develop the lateral surface of the cylinder of 40mm diameter and height 60mm cut in the following way.

Fig 1B

Fig 2B

2A
D=3.14x40=125.6mm
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 59

45) Develop the lateral surface of the cylinder of 40mm diameter and height 60mm which is cut in the
following way.

Fig 1B

Fig 2B
Note:Development should be started about EE1 and not
AA1,else It will not be a single piece development

Fig 1A 2A
D=3.14x40=125.6mm
4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 60
46) A pipe made of using a half tubular (circular) with a half square in shape is cut as shown in the following
figure. Draw the development of the lateral surface of the object.

Fig 1B

Fig 2B

2A Perimeter of top view

D/2=3.14x50/2=78.54mm+90
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 61

47) A cone of base diameter 60mm and 70mm is resting on its base on HP. It is cut as shown in the following
figure. Draw the development of the lateral surface of the remaining portion of the cone.

Fig 2B
Fig 1B

Fig 1A 2A :
= 3600 *r / R
= 3600 * 30/76.16=141.80

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 62

48) A cone of base diameter 60mm and height 70mm is resting on its base on HP. It is cut as shown in the
following figure. Draw the development of the lateral surface of the remaining portion of the cone.

Fig 2B
Fig 1B

Fig 1A 2A
= 3600 *r / R
= 3600 * 30/76.16=141.80
4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 63
49) A right cone of 55mm diameter of base and 75mm height stands on its base on HP. It is cut to the shape of
a truncated cone with its truncated surface inclined at 450 to the axis lying at a distance of 40mm from the
apex of the cone. Obtain the development of the lateral surface of the truncated cone.

Fig 2
Fig 1B

= 3600 *r / R
= 3600 * 27.5/79.88=135.20

Note:Development should be preferably started about OE

and not OA.

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 64

50) Draw the development of the following truncated cone.

Fig 1B
Fig 2B

2A
= 3600 *r / R
Fig 1A = 103.440
4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 65
 51. A frustum of a cone 80mm bottom diameter and 40mm top diameter, and 70mm high
rest on base on HP. It is cut through by a semicircular hole of diameter 40mm as in fig. 9 Draw
the development.

Fig 1B Fig 2B

72,8
A

126,88
124,8
K
145,6

O 40 9).

,6
145
I
125
70

R2

127
0

3' 4'
G

72,8
2'
1' 5'
a' b' c' d' e' f' g' F
O 80
j

99°
k i 5
E
4
l h N

4°
3
D
2 M

8°
a g

Fig 2A 1 C

Fig 1A b f
=(r/R)*360=98.90
B
=(40/145.6) X 360=98.90
e
c m n
d
A
4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 66
52) Draw the development of the lateral surface of the cone, whose front view is as shown in the following
figure.

Fig 1B

Fig 2B

2A
= 3600 *r / R
= 3600 * 25/65=138.460

Fig 1A
4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 67
 53 A frustum of a cone 80mm bottom diameter and 40mm top diameter, and 70mm high
rest on base on HP. It is cut through by a semicircular hole of diameter 40mm as in fig. Draw
the development.

Fig 2B
Fig 1B

72,8
A

126,88
124,8
K
145,6

O 40 9).

,6
145
I
125
70

R2

7
0

12
3' 4'
G

72,8
2'
1' 5'
a' b' c' d' e' f' g' F
O 80
j

99°
k i 5
E
4
l h N

4°
3
D
2 M

8°
a g

Fig 2A 1 C

b f
=(r/R)*360=98.90
B
=(40/145.6) X 360=98.90
e
c m n
d
A
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 68

 54. A cone 40 mm diameter of base and height 50 mm rests with its base on HP. An isosceles
triangular through slot having its axis perpendicular to VP and parallel to HP is cut through
the cone such that the base of the slot is parallel to the base of the cone and 10 mm above it,
and one of the sides of the slot is parallel to one of the end generators of the cone and the
other side coincides with the other end generator of the cone, with the apex of the slot 15
mm from the apex of the cone. front views, and development.
A
85
53, L
Fig 1B

21,54
17,32

32,31

43,08
K

16,16

,31
R 16,16
32
15

R
5
6
4
J
7

°
3

134
50

I This is a curve
8
,08
2 43
R
9
1
H
10

G
Fig 2B
k j F
i
l E
h D
B C
A
g
a R 20
f
Fig 2A
b
3,0 e =(r/R)*360=133.70
3 c d
=(20/53.85) X 360=133.70

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 69

 56. A cone 40 mm diameter of base and height 50 mm rests with its base on HP. An isosceles
triangular through slot having its axis perpendicular to VP and parallel to HP is cut through the cone
such that the base of the slot is parallel to the base of the cone and 10 mm above it, and both the
sides of the slot are parallel to end generators of the cone, with the apex of the slot 15 mm from
the apex of the cone. Show the shape of the slot in the top and front views, and development.

Fig 1B A
85 Fig 2B
53, L

K
16,16
15

32,31

16
43,0

16,
53,8
J

R
8

°
134
50

8 I This is a curve
3,0

,31
4

R 32
R
H
10

k j F
i
l E
h D
B C
A
g
a R 20
16

b
f Fig 2A
e =(r/R)*360=133.70
c d
3,03 =(20/53.85) X 360=133.70

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 70

57) A cone of base diameter 50mm and height 60mm is resting with its base on HP. It is cut, as shown in the
following front view of which is as shown in figure, Draw the development of the lateral surface of it.

Fig 2B
Fig 1B

Fig 2A
= 3600 *r / R
= 3600 * 25/65=138.460
Fig 1A
4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 71
58) Draw the developments of the lateral surface of a funnel consisting of a cylinder and a frustum of a cone.
The diameter of the cylinder is 20mm and top face diameter of the funnel is 80mm. The height of frustum and
cylinder are equal to 60mm and 40mm respectively.
3A
Fig 1B
= 3600 *r / R
= 3600 * 40/89.44=1610

Fig 2B

2A Fig 3B
D=3.14x20=62.83mm

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 72

59) Draw the development of the lateral surface of the cut cone, whose front view is shown in the following
figure.

Fig 1B

Fig 2B

Fig 2A
= 3600 *r / R
= 3600 * 25/65=138.460

Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 73

60) A funnel is to be made of sheet metal. The funnel tapers from 40mm to 20mm diameter to a height of
20mm and from 20mm to 15mm diameter, for the next 20mm height. The bottom of the funnel is bevelled off
to a plane inclined at 450 to the axis, Draw the development of the funnel.

2A 3A
= 3600 *r / R = 3600 *r / R
= 3600 * 20/44.72=1610 = 3600 * 10/ 80.62=44.650
Fig 1B

Fig 2B

Fig 3B
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 74

61) A funnel is made of sheet metal. The funnel tapers from 60mm to 30mm diameters to a height 50mm. And
then forms to a cylinder with a height of 50mm. Bottom of funnel is beveled off completely at an angle of 450 to
axis Draw the development of funnel.
2A Fig 1B
= 3600 *r / R
= 3600 * 30/62.66=172.540
Fig 3B

Fig 2B

3A
D=3.14x30=94.25 mm
Fig 1A

4-May-20 DEPARTMENT OF MECHANICAL ENGINEERING, BMSCE, BANGALORE-19 75