UNIT III

Projection of Solids

Solids

1. A solid is a three dimensional object, havinglength, breath and thickness.

2. It is bounded by plane faces or curvedsurfaces or combination of plane and curvedareas.

Solids

1. Types of solids

a) Polyhedra

b) Solids of revolution

Polyhedra

1. A polyhedron is a many sided solid figurebounded by only plane surfaces or faces.

2. If all the faces of a polyhedron are having thesame size and shape, it is said to be a regularpolyhedron.

Polyhedra Examples

Cube

Tetrahedron

Octahedron

Dodecahedron

Polyhedra

There are polyhedra other than regular oneswhich are called prisms and pyramids.

1. Prism

a) A prism is a polyhedron having two equal andsimilar end faces called top face and bottom facejoined by other faces which may be rectangles orparallelograms.

b) The imaginary line joining the centres of thefaces is called the Axis.

Prisms Examples

Polyhedra

1. Pyramids

a) A pyramid is a polyhedron having a plane figurefor its base and equal number of isoscelestriangular faces meeting at a point called vertexor apex.

Pyramids Examples

Polyhedra

2. Solids of revolution

If a plane surface is revolved about one of itsedges, the solid generated is called a solid ofrevolution.

Solids of revolution

Other forms of solids

Frustum

When a pyramid of a cone is cut by acutting plane parallel to its base, the remainingportion thus obtained after removing the topportion is called the Frustum.

Frustum

Frustum of Pyramid and Cone

Other forms of solids

Truncated

When a solid(prism/cylinder/pyramid/cone) is cut by a cutting plane inclined to its base (not parallel) the remaining portion thus obtained after removing the top portion is called the truncated solid.

Truncated

Truncated Pyramid and Cone

Solids with Axis inclined to the VP and parallel to the HP

1. In first stage, it is assumed to be kept suchthat the axis is perpendicular to the VP.

2. In this simple position, the front view isdrawn first and the top view projected fromit.

3. In the second stage, the projections areobtained by any one of the method

a) Change of position method

b) Change of reference line or Auxiliary projectionmethod

Solids with Axis inclined to the HP and parallel to the VP

1. In first stage, it is assumed to be kept suchthat the axis is perpendicular to the HP.

2. In this simple position, the top view is drawnfirst and the front view projected from it.

3. In the second stage, the projections areobtained by any one of the method

a) Change of position method

b) Change of reference line or Auxiliary projectionmethod

Change of position method

Problem 1

A cube of side 40 mm rests on the groundwith a face parallel to VP and 25 mm in frontof VP. Draw the top and front views of thecube.

V.P.X Y

H.P.

Problem 1

p

r

q

s

(t) (u)

(v)(w)

40

(p’)

(t’)

(q’)s’

(u’)w’

r’

v’

40

25

Problem 2

A cube of side 40 mm rests on the HP on oneof its ends with a vertical face inclined at 40to the VP. Draw its projections.

V.P.X Y

H.P.

Problem 2 Draw the top view as a square of side 40 mm with one of its sides inclined at 40 to xy.

=40

p r

q

s

(t)

(u)

(v)

(w)

40

p’

t’

q’ (s’)

u’

(w’)

r’

v’

40

Problem 3

Draw the projections of a pentagonal prism ,base 25 mm side and axis 50 mm long,resting on one of its rectangular faces on theH.P. with the axis inclined at 45 to the V.P.

X Y

1’

2’

3’

(d’)5’25

50

a b c(d)(e)

1 2 3(5) (4)

a

b

cd

e

1

2

3

5

4

a1’

b1’

c1’

d1’e1’

11’

21’

31’

41’51’45

Problem 3

(a’)

(b’)

(c’)

4’(e’)

Problem 4

Draw the projections of a hexagonal prism ofbase side 20 mm and axis length 50 mmwhen it is lying on the ground on one of itsrectangular faces and the axis is inclined at35 to the VP.

Problem 4

V.P.X Y

H.P.

20

p’

q’ r’

s’

t’u’

(1’)

(2’) (3’)

(4’)

(5’)(6’)

50

p q r s(t)(u)

1 2 3 4(5)(6)

p1

s1

(t1)

(u1)

3511

21

31

41

(51)

(61)

q1

r1

p1’

q1’ r1’

s1’

t1’u1’

(11’)

21’ 31’

41’

51’61’

Problem 5

A hexagonal prism of base side 30 mm andaxis length 60 mm rests on the HP on one ofits base edges with its axis inclined at 60 tothe HP and parallel to the VP. Draw its topand front views.

Problem 5

Problem 6

A hexagonal prism of base side 25 mm andaxis length 55 mm rests on the HP on one ofits base corners such that a solid diagonalpassing through that corner is perpendicularto the HP. Draw its projections.

Problem 6

Problem 7

Draw the projections of a cylinder ofdiameter 50 mm and axis length 80 mmwhen it is lying on the ground with its axisinclined at 45 to the VP and parallel to theground.

Problem 7

V.P.X Y

H.P.

p’

q’

r’

s’

t’

u’w’v’

80

p q r s t

1 2 3 4 5 =45

(1’)

(2’)

(3’)

(4’)

(5’)

(u)(v)(w)

(6)(7)(8)

(6’)(8’)

(7’)11

51p1

t1

21

q1

31

r1

41

s1

(61)

(71)

(81)

(u1)

(v1)

(w1)

p’1 t’1

v’1

r’1 3’1

5’1

7’1

(1’1)50

Problem 8

A cylinder of diameter 30 mm and axis length50 mm is resting on the HP on a point so thatits axis is inclined at 45 to the HP andparallel to the VP. Draw its top and frontviews.

Problem 8

H.P.X Y

V.P.

p

q

rs

t

uwv

50

p’ q’ r’ s’ t’

1’ 2’ 3’ 4’ 5’

=45

(1)

(2)

(3)

(4)

(5)

(u’)(v’)(w’)

(6’)(7’)(8’)

(6)(8)

(7)

p1t1

1’1

r131

(51)

71

11

5’1

3’1(7’1)

2’1(8’1)

4’1(6’1)

p’1q’1(w’1)

r’1(v’1)

s’1(u’1)

t’1

21 (41)

(61)81

v1

u1

q1 s1

w1

30

Problem 9

A cone of base 40 mm diameter and axis 50mm long touches VP on a point of its basecircle. Its axis is inclined at 30 to VP andparallel to HP. Draw its projections.

Problem 9

V.P.X Y

H.P.

p’

q’

r’

s’

t’

u’w’v’

40

50

p q r s t

o’

(u)(v)(w)

o

p

q

r

s

t(u)

(v)

(w)

o

30

p’1

q’1

r’1

s’1

(t’1)

u’1

v’1

w’1

Problem 10

A cone of base diameter 40 mm and altitude80 mm rests on the HP with its axis inclinedat 30 to the HP and parallel to the VP. Drawits front and top views.

Problem 10

V.P.X Y

H.P.

p

q

rs

t

uwv

40 o

80

t’u’v’w’p’ (q’) (r’) (s’)

o’

p1’

30

p1

q1

r1

s1

(t1)

u1

v1

w1

(q1’)w1’

(r1’)v1’

(s1’)u1’

t1

o1’

(o1)

Problem 11

A cone of base diameter 40 mm and height56 mm is freely suspended from one of itsbase points such that its axis is parallel to theVP. Draw its projections.

Problem 11

V.P.X Y

H.P.

1

8

76

5

42

3

40 o

56

5’4’3’2’1’ (8’) (7’) (6’)

o’

14

C.G

Mark the centre of gravity (C.G) of the cone at 1/4th

height (14 mm) from the base in the front view.

Let the cone be freely suspended from the point 5.

The axis of the cone will be parallel to VP if the cone is suspended such that the line 5’ – C.G is perpendicular to XY.Tilt the front view such that 5’-C.G is perpendicular to XY. This tilted front view is the final front view of the suspended cone.

C.G

51’

11’

(71’)31’

o1’

(61’)41’

(81’)21’

o1 11

21

31

41

51

61

71

81

Problem 12

A cone of base diameter 42 mm and axislength 65 mm rests on the HP on a point inthe circumference of the base with one of itsslant generators perpendicular to the HP andparallel to the VP. Draw its projections.

Problem 12

Problem 13

A hexagonal pyramid of base edge 40 mmand altitude 80 mm rests on one of its baseedges on the HP with its axis inclined at 30to the HP and parallel to the V.P. Draw its topand front views.

X Y

a

b

c

d

e

f

40

(a’) b’ c’ (f’) (e’)d’

o’

60º

e1’(d1’)

(a1’)b1’

c1’ (f1’)

o1’

30º

a1

b1

c1

d1

e1

f1

o1o

Problem 13

V.P.

H.P.

80

Problem 14

A hexagonal pyramid of base side 24 mm andaxis length 40 mm is lying on the HP on oneof its triangular faces with its axis parallel tothe VP. Draw the top and front views of thepyramid.

Problem 14

Problem 15

Draw the projections of a pentagonalpyramid of base 25 mm side and axis 60 mmlong when it is lying on the HP on one of itsbase edges, such that the axis is parallel tothe VP and inclined at 30 to HP.

Problem 15

V.P.X Y

H.P.

25

p

t

s

r

q

o

o’

60

p’ r’(s’)q’(t’)

o’

p’1

r’1(s’1)

q’1(t’1)

30

p1

q1

(r1)

(s1)

t1

o1

Problem 15A

A pentagonal pyramid side of base 20 mmand axis 45 mm long rests with one of itscorners on HP such that the base is inclinedat an angle of 60° to HP and one side of baseis perpendicular to VP. Draw its projections.

Problem 15A

Problem 16

A pentagonal pyramid of base edge 25 mmand axis length 60 mm rests on one base sideon HP such that the highest base corner is 20mm above HP. Its axis is parallel to the VP.Draw its top and front views.

Problem 16

Problem 17

A pentagonal pyramid of base side 25 mmhas an altitude of 45 mm. The pyramid restson the HP on one of its base sides such thatthe triangular face containing that side isperpendicular to both the HP and the VP.Draw its projections.

Problem 17

Problem 18

A square pyramid of base side 30 mm axislength 50 mm has one of its triangular facesin the VP and the axis parallel to and 25 mmabove the HP. Draw its projections.

Problem 18

V.P.X Y

H.P.

30o’

p’ s’

r’q’

25

50

q(p) r(s)

o

r1(s1) o1

q1(p1)

S1’

p1’

o1’

q1’r1’

Make r1 as centre and cut an arc at a distance of rq

Make o1 as centre and cut an arc at a distance of oq

Problem 19

A square pyramid of base side 60 mm andaltitude 100 mm lies on the HP on one of itstriangular faces with its axis parallel to theVP. Draw its projections. Page no 217

Problem 19

V.P.X Y

H.P.

60o

p q

rs

100

o’

q’(r’)p’(s’)

p1’(s1’)

q1’(r1’) o1’

o1

s1r1

p1q1

This triangular face will be tilted to lie on the HP in the next stage

Problem 20

A square pyramid of base side 30 mm andheight 50 mm rests on the ground on one ofits base edges such that its axis is inclined at45 to the ground and parallel to VP. Draw itsprojections.(univ Question)

Problem 21

A square prism of base side 35 mm and axislength 60 mm rests on one of its base edgeson the HP with its axis inclined at 30° to theHP and parallel to the VP. Draw its top andfront views.

Problem 21

Problem 22

Draw the projections of a cube of side 40 mmwhen it rests on the ground on one of itscorners and a face containing that corner isinclined at 30° to the ground andperpendicular to the VP.

V.P.X Y

H.P.45°45°

p’ s’(q’) r’

t’w’(u’)

v’

p(t)

q(u)

s(w)

r(v)

40

p’1

s’1(q’1)

r’1

v’1

w’1(u’1)t’1

Problem 22

p1

q1

r1

s1

(t1 )

u1

v1

w1

30°

40

Problem 23

Draw the projections of a cube of side 40 mmwhen it rests on one of its corners with adiagonal of the solid vertical.

V.P.X Y

H.P.45°45°

p’ s’(q’)r’

t’w’(u’)

v’

p(t)

q(u)

s(w)

r(v)

40

p’1

s’1(q’1)

r’1

v’1

w’1(u’1)

t’1

Problem 23

p1

q1

r1

s1

(v1 )

u1

t1

w1

40