UNIT - IV

DEVELOPMENT OFLATERAL SURFACES OF SIMPLE

AND SECTIONED SOLIDS

Definitions

Development of Surfaces of the solid :

1. Suppose an object like a square prism iswrapped around by using paper.

2. When the wrapper is opened and spreadout on a plane surface, the resulting figureis called the development of the surfacesof the solid.

Note :

The development of any solid shows thetrue shape of all the surfaces of the solid.

Methods of Development

1. Parallel line method

2. Radial line method

3. Approximate method

Note :

1. The development of the lateral surfaces ofthe objects only are shown.

2. The base and top are cut to the requiredgeometrical shape and fastened suitably.

PRISMS & CYLINDERS –Parallel Line Development Method

http://www.youtube.com/watch?v=IwlrJOHgOB8

Development of Prism

End

Lateral Sides

Lateral Surface

Problem 1

Draw the development of the lateral surfacesof a right square prism of edge of base 30mm and axis 50 mm long.

Problem 1

50

30

a (a1) b (b1)

c (c1)d (d1)

a’1 (d’1) b’1 (c’1)

b’(c’)a’(d’)

Draw a line equal to the length of the perimeter of the base of the prism.

120

A1 B1 C1 D1 A1

A B C D A

Development of Cylinder

End

Lateral Sides

Lateral Surface

Problem 2

Draw the development of the completesurface of a cylindrical drum. Diameter is 40mm and height 60 mm.

Problem 2

60

p’ q’ r’ s’ t’

a’e’

p(a)

q(b)

r(c)s(d)

t(e)

(u’)(v’)(w’)

d’(f’)c’(g’)

b’(h’)

u(f)w(h)

v(g)

40

Draw a line equal to the length of the circumference of the base circle (D).

125.6

A B C D

P Q R S T U V W P

E F G H A

Problem 3

A hexagonal prism, edge of base 20 mm andaxis 50 mm long, rests with its base on HPsuch that one of its rectangular faces isparallel to VP. It is cut by a planeperpendicular to VP, inclined at 45° to HP andpassing through the right corner of the topface of the prism.

(i) Draw the sectional top view.

(ii)Develop the lateral surfaces of thetruncated prism.

Problem 3

1’

4’

3’

2’50

a(a1)

45

20

(5’)

(6’)

45

b(b1) c(c1)

d(d1)

e(e1)f(f1)

a’1 d’1b’1(f’1) c’1(e’1)

a’ b’(f’) c’(e’) d’

20

A1

A

B1

B C D E F A

C1 D1 E1 F1 A1

1

2

3 4 5

6

1

Draw six equal rectangles to represent the development of the lateral surfaces of the hexagonal prism in thin lines.

Problem 4

A hexagonal prism of base side 20 mm andheight 45 mm is resting on one of its ends onthe HP with two of its lateral faces parallel tothe VP. It is cut by a plane perpendicular tothe VP and inclined at 30° to the HP. Theplane meets the axis at a distance of 20 mmabove the base. Draw the development ofthe lateral surfaces of the lower portion ofthe prism. Page no 300

Problem 4

Problem 5

A pentagonal prism, side of base 25 mm andaltitude 50 mm, rests on its base on the HPsuch that an edge of the base is parallel to VPand nearer to the observer. It is cut by aplane inclined at 45° to HP, perpendicular toVP and passing through the center of theaxis.

(i) Draw the development of the completesurfaces of the truncated prism.

Problem 5

50

3’

1

1’

2’

45°

3

4

5

25

a(a1)

b(b1) c(c1)

d(d1)

e(e1)a’1 d’1b’1 c’1(e’1)

a’ b’ (e’) c’ d’

2

(5’)

4’

25 1

A1

A

B1

B

C1

C

D1

D

E1

E

A1

A

2

3

4

5

1

Problem 6

A pentagonal prism of side of base 30 mmand altitude 60 mm stands on its base on HPsuch that a vertical face is parallel to VP andaway from observer. It is cut by a planeperpendicular to VP, inclined at an angle of50° to HP and passing through the axis 35mm above the base. Draw the developmentof the lower portion of the prism. Page no15.4 (Exercise)

Problem 7

Draw the development of the lateral surfaceof the lower portion of a cylinder of diameter50 mm and axis 70 mm when sectioned by aplane inclined at 40° to HP and perpendicularto VP and bisecting axis.

Problem 7

70

p’ q’ r’ s’ t’

a’e’

p(a)

q(b)

r(c)s(d)

t(e)

(u’)(v’)(w’)

d’(f’)c’(g’)

b’(h’)

u(f)w(h)

v(g)

50

Draw a line equal to the length of the circumference of the base circle (D).

157

A B C D

P Q R S T U V W P

E F G H A

351’

40°

2’

3’

4’5’

(6’)

(7’)

(8’)

1

23

4

5

67

8

1

2

3

45

6

7

8

1

Problem 8

A cylinder of diameter 40 mm and height 50mm is resting vertically on one of its ends onthe HP. It is cut by a plane perpendicular tothe VP and inclined at 30° to the HP. Theplane meets the axis at a point 30 mm fromthe base. Draw the development of thelateral surface of the lower portion of thetruncated cylinder. Page no 305

Problem 8

Problem 9

A cylinder of diameter 40 mm, height 75 mmis cut by a plane perpendicular to VP andinclined at 55° to HP meeting the axis at topface. Draw the lateral development of thesolid.

PYRAMIDS & CONES –Radial Line Development Method

Development of Pyramid

Problem 10

Draw the development of the lateral surfacesof a square pyramid, side of base 25 mm andheight 50 mm, resting with its base on HPand an edge of the base parallel to VP.

Problem 10

a

o

25

b

cd

50

a’(d’) b’(c’)

o’

If the top view of a slant edge of a pyramid is parallel to XY, then the front view of that edge will give its true length.

To obtain the true length of a slant edge make “ob” parallel to XY.O as center and ob as radius draw an arc to cut the horizontal drawn from o at a1.

a1

a’1

True length of slant edge

O

A

B

CD

A25

Development of Cone

End

Circumference of base circle

Slant length

Problem 11

Draw the development of the lateral surfaceof a cone of base diameter 48 mm andaltitude 55 mm.

Problem 11

ao

48

b

c

d

55

a’ c’

o’

True length (L) of slant generator O

A

B

CD

A

b’(d’)

144°

= (Base circle radius/True slant length)*360°

= (24/60)*360°

=144°

Divide 144° into 4 equal parts.Per division 36°

36°

Problem 12

A square pyramid of base side 25 mm andaltitude 50 mm rests on it base on the HPwith two sides of the base parallel to the VP.It is cut by a plane bisecting the axis andinclined a 30° to the base. Draw thedevelopment of the lateral surfaces of thelower part of the cut pyramid.

Problem 12

a

o

25

b

cd

50

a’(d’) b’(c’)

o’

o’a’1 gives the true length of the slant edge.

To obtain the true length of a slant edge make “ob” parallel to XY.O as center and ob as radius draw an arc to cut the horizontal drawn from o at a1.

a1

a’1

True length of slant edge

O

A

B

C

D

A

25

30°

1’

2’(3’)

(4’)1’’(4’’)

2’’(3’’)

O1=O4=o’1” , similarly O2=O3=o’2”

1 2

3 4

1

Problem 13

A square pyramid base 35 mm side axis 70mm long rests on its base on HP such thattwo adjacent sides of the base are equallyinclined to VP. It is sectioned by a planeperpendicular to VP inclined a 30° to HP andpassing through the mid-point of the axis.Draw the sectional top view and develop thelateral surfaces of the truncated pyramid.Page no 15.12

Problem 13

Problem 14

A regular hexagonal pyramid of side of base30 mm and height 60 mm is resting verticallyon its base on HP such that two of the sidesof the base are perpendicular to VP. It is cutby a plane inclined at 40° to HP andperpendicular to VP. The cutting plane bisectsthe axis of the pyramid. Obtain thedevelopment of the lateral surface of thetruncated pyramid.

Problem 14

p 2

q

s

p’(u’) q’(t’)

60

3’(4’)

r’(s’)

1

1’(6’)

2’(5’)

r

30

40°

3

5

Sectional top view

30o

o’

a’

a

t

u

46a1

a’1

1’’(6’’)

2’’(5’’)

3’’(4’’)

O

P

1

QR

S

T

U

P

2

3 4

5

6

1

Problem 15

A hexagonal pyramid of base of side 25 mmand altitude 50 mm is resting vertically on itsbase on the ground with two of the sides ofthe base perpendicular to the VP. It is cut bya plane perpendicular to the VP and inclinedat 40° to the HP, The plane bisects the axis ofthe pyramid. Draw the development of thelateral surfaces of the pyramid. Page no 313

Problem 15

Problem 16

A pentagonal pyramid side of base 30 mmand height 52 mm stands with its base on HPand an edge of the base is parallel to VP andnearer to it. It is cut by a plane perpendicularto VP, inclined at 40° to HP and passingthrough a point on the axis 32 mm above thebase. Draw the sectional top view. Developthe lateral surface of the truncated pyramid.Page no 15.12

Problem 16

30

a b

c

de

o

a’ (e’) b’ c’(d’)

52

o’

32

40

1’

2’

3’(4’)

(5’)

1

2

3

45

a1

a’1

O

1

BC

D

E

A

2

3

4 5

6

A

1”

2”

3”4”

5”

Problem 17

A pentagonal pyramid of base side 25 mmand height 60 mm is resting vertically on itsbase on the ground with one of the sides ofthe base parallel to the VP. It is cut by a planeperpendicular to the VP and parallel to theHP at a distance of 25 mm above the base.Draw the development of the lateral surfacesof the frustum of the pyramid . Also show thetop view of the cut surface. Page no 311

Problem 17

Problem 18

A Cone of base diameter 60 mm and height70 mm is resting on its base on HP. It is cut bya plane perpendicular to VP and inclined at30° to HP. The plane bisects the axis of thecone. Draw the development of its lateralsurface.

Problem 18

V.P.X Y

H.P.

70

o’

a’ e’

30

a

b

cd

e

d’(f’)c’(g’)

b’(h’)

fh

g

60

351’2’(8’)

3’(7’)

4’(6’)

o

5’

1”

3”

2”

4”5”

(6”)(7”)

(8”)

oa is parallel to XY. So o’a’ = OA = True length of the generator (slant height)

O

A

B

C

D E

142°

= (r/L)*360°

= (30/76)*360°

=142°Divide 142° into 8 equal parts.Per division 17.75°

17.8°

F

G

H

A

1

23 4 5 6 7

8

1

Problem 19

A Cone of base 50 mm diameter and 60 mmheight, rests with its base on HP. It is cut by asection plane perpendicular to VP, parallel toone of the generators and passing through apoint on the axis at a distance of 22 mm fromthe apex. Draw the sectional top view anddevelop the lateral surface of the remainingportion of the cone. Page no 15.23

Problem 19

V.P.X Y

H.P.

60

o’

p’ t’

a

b

cd

e

s’(u’)r’(v’)q’(w’)

fh

g

50

22

1’(7’)

2’(6’)

3’(5’)4’

1

23

56

7

o 4

a’

a

2”(6”)

3”(5”)

4”

O

A

B

C

DE

138.5°17.3°

F

G

H

A

1

2

34

5

6

7

Measure in top view b to 1.b1=B1 ; g7=G7

DEVELOPMENT OFLATERAL SURFACES OF SOLIDS WITH CUT-OUTS AND HOLES

Problem 20

A Square prism of 36 mm edge of base and64 mm height stands on HP with two of itsbase edges equally inclined to VP. It has asquare hole of 24 mm side centrally cut rightthrough the prism such that its faces areequally inclined to HP. Axis of the hole isparallel to HP and perpendicular to VP. Drawthe development of the lateral surfaces ofthe prism showing the true shape of thesquare cutout formed it. Page no 16.1

Problem 20

a

36

b

c

d

64

a’

c’b’(d’)

32

24

1’(11’)

2’(21’)

3’(31’)

4’(41’)

mn

m1 n1

36 X 4

A B C D A

AM=am CN = cn

M

1

2

3

4

31

21

11

41

N

AM1 = am1 CN1 = cn1

N1 M1

Problem 21

A hexagonal prism of side of base 24 mm andaxis 64 mm is on HP on one of its ends with abase edge parallel to VP. A square hole ofside 26 mm is drilled such that the axis ofthe hole is perpendicular to VP an bisects theaxis of the prism with all the faces equallyinclined to HP. Develop lateral surfaces. pageno 16.2

Problem 21

Problem 22

A Pentagonal prism of side of base 25 mmand axis 60mm is on HP on one of its endswith a base edge parallel to VP and nearer toit. A square hole of side 25 mm is drilled suchthat axis of the hole is perpendicular to VPand bisects axis of the prism with all thefaces equally inclined to HP. Draw thedevelopment of lateral surfaces of the prismshowing true shape of the hole on it. Page no16.2

Problem 22

Problem 23

A Cylinder of 50 mm base diameter and axis70 mm long rests on its base on HP. A squarecutout of 35 mm side is drilled through thecylinder such that axis of cutout isperpendicular to the axis of the cylinder. Thecenter of the cutout is 35 mm above HP and15 mm away from the axis of cylinder. Twofaces of the cutout are equally inclined to HP.Develop lateral surfaces. Page no 16.3

Problem 23

Problem 24

A hexagonal prism of side of base 25 mm andaltitude 65 mm rests on its base on HP,having a rectangular face of the prismparallel to VP. A horizontal hole of 35 mmdiameter is centrally drilled in it, such thatthe axis of the hole is normal to VP. Developthe lateral surfaces of the prism with theshape of hole.

Problem 24

1’

2’

65

a(a1)

25

3’

4’

b(b1) c(c1)

d(d1)

e(e1)f(f1)

a’1 d’1b’1(f’1) c’1(e’1)

a’ b’(f’) c’(e’) d’

25

A1

A

B1

B C D E F A

C1 D1 E1 F1 A1

1

P

3

4

56

32.5

35

p

ap = A1P CN = cn

5’6’

q

2

Q

Since the hole portion 65 and 23 are on the face BCB1C1 which has true shape in the front view also, they will appear in the development as they appear in the front view.

O

The curves 65 and 23 are circular and can be drawn with O as centre and 17.5 mm as radius.

O1

11

3121

41

5161

Problem 25

A pentagonal pyramid of side of base 24 mmand axis 60 mm long stands on its base on HPwith a side of base parallel to VP and nearerto it. A square cutout of 15 mm side is drilledthrough it such that its axis is parallel to HPand perpendicular to VP. Axis of cutout meetsthe axis of the pyramid 15 mm from base.Faces of the cutout are equally inclined to HP.Develop the lateral surfaces. Page no 16.8

Problem 25

24

a

b c

d

e

o

a’ (b’) e’ d’(c’)

60

o’

15

1’

3’

(8’)

(2’)

p1

q

q1

p

a1

a’1

O

1

ED

C

B

A

7

3

5

A

(4’)

(6’)

5’

7’

Join the apex to the extreme points of the square hole to meet the base, i.e.,join o’1’. Extend it to meet the base of the pyramid at p’. Similarly mark q’.

p’ q’

1’1, 5’1

2’1, 4’1

3’1

8’1, 6’1

7’1

True length of slant edge

P

P1

AP = ap ; AP1 = ap1

QQ1

3

5

7

4

6

2

8

1