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ENGINEERING GRAPHICS AND COMPUTER AIDED DESIGN

Module-II: Projection of Solids
Projections of solids. Solids in simple positions and axis inclined to one plane only. Section
of solids. Section planes inclined to Horizontal Plane only. True shape of the section.
(Manual and CAD Drawing)

A solid is a three-dimensional object having length, breadth and thickness. It is bounded by plane
faces or curved surfaces or combination of plane and curved areas. Various types of solids are
used in engineering practice. In general, all these solid objects are broadly categorized as:
1. Polyhedra
2. Solids of revolution
Polyhedra:
A polyhedron is a three-dimensional solid bounded by flat polygonal faces, straight edges, and
sharp corners. The plural of a polyhedron is called as polyhedra.
Regular Polyhedra: If all the faces of a polyhedron are having the same size and shape, it is said
to be a regular polyhedron. Examples: Cube, Tetrahedron, Octahedron, Dodecagon, Icosahedron.

When faces of a polyhedron are not formed by equal identical faces, they may be classified into
prisms and pyramids
Prism: A prism is a polyhedron having two equal and similar end faces called top face and bottom
face joined by other faces which are normally rectangles. The imaginary line joining the centres
of the faces is called the axis.

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Prism - Examples
Pyramid: A pyramid is a polyhedron having a plane figure for its base and equal number of
isosceles triangular faces meeting at a point called vertex or apex.

Pyramid - Examples

Solids of Revolution: A solid of revolution is a solid generated rotating a plane area about its
axis. Following are the solids of revolution.
1. Cylinder
2. Cone
3. Sphere

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Cylinder Cone Sphere

ORTHOGRAPHIC PROJECTION OF SOLIDS IN DIFFERENT POSITIONS

The position of a solid object in space or resting on the reference planes may be specified by
considering the location of its axis, base, corner, edge, diagonal or surfaces, with the reference
planes. Following are some of the positions of the solids considered for this course.
1. Axis perpendicular to HP and parallel to VP
2. Axis perpendicular to VP and parallel to HP
3. Axis Parallel to HP and VP
4. Axis inclined to HP and parallel to VP
5. Axis inclined to VP and parallel to HP

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Example Problems - Axis perpendicular to HP and parallel to VP

Problem 1: A square prism of side 35 mm and axis 60 mm is resting on its base on HP such that
the two base edges are parallel to VP. Draw its projections.
Steps: (i) Draw the top view first, since it shows the true shape of the base of the solid.
(ii) Project the front from the top view.

a’ b’ c’ d’

60

p’ q’ r’ s’
X Y

a p s d

b q r c All Dimensions are in mm

35

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 2
A square prism of side 35 mm and axis 80 mm is resting on its base on HP such that, two base
edges are equally inclined to VP. Draw its projections.
Steps: (i) Draw the top view first, since it shows the true shape of the base of the solid.
(ii) Project the front from the top view.

a’ b’ d’ c’

80

p’ q’ s’ r’
X Y

d
450 s 450

a p r c

q 35
b

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 3
A triangular prism of side 25 mm and axis length of 40 mm is resting on HP with a face parallel
to VP. Draw the projections.
Steps: (i) Draw the top view first, since it shows the true shape of the base of the solid.
(ii) Project the front from the top view.

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 4
A cone of base diameter 35 mm and axis length 55 mm is resting on HP on its base. Draw the
projections.
Steps: (i) Draw the top view first, since it shows the true shape of the base of the solid.
(ii) Project the front from the top view.

o’

55

X Y

Ф35
All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 5
A cylinder of base diameter 35 mm and axis length 55 mm is resting on HP on its base. Draw the
projections.
Steps: (i) Draw the top view first, since it shows the true shape of the base of the solid.
(ii) Project the front from the top view.

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 6
A hexagonal pyramid of base side 25 mm and axis length 55 mm is resting on HP on its base with
two of its base edges parallel to VP. Draw the projections.
Steps: (i) Draw the top view first, since it shows the true shape of the base of the solid.
(ii) Project the front from the top view.

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 7
A pentagonal prism of side 35 mm and axis 70 mm is resting on its base on HP, such that, one of
its base edges is parallel and nearer to VP. Draw its projections.
Steps: (i) Draw the top view first, since it shows the true shape of the base of the solid.
(ii) Project the front from the top view.

b’ c’
a’
e’ d’

70

t’ s’
X Y
p’ q’ r’

e d
t s

a p r c

q
35
b

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Example Problems - Axis perpendicular to VP and parallel to HP

Problem 8: A triangular prism of base side 35 mm and axis 60 mm is resting on HP on one of its
rectangular faces, with its axis perpendicular to VP. Draw its projections.
Steps: (i) Draw the front view first, since it shows the true shape of the base of the solid.
(ii) Project the top from the front view.

c’

r’ 35

a’ p’ q’ b’
X Y

p r q

60

a c b

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 9: A cylinder of base diameter 35 mm and axis length 60 mm is resting on HP on one of its
generators, with its axis perpendicular to VP. Draw its projections.

Steps: (i) Draw the front view first, since it shows the true shape of the base of the solid.
(ii) Project the top from the front view.

Ф35

X Y

60

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 10: A square prism of base side 35 mm and axis 60 mm is resting on HP on one of its
rectangular faces, with its axis perpendicular to VP. Draw its projections.
Steps: (i) Draw the front view first, since it shows the true shape of the base of the solid.
(ii) Project the top from the front view.

d’ s’ r’ c’

35

a’ p’ q’ b’
X Y
’

p s Q r

60

a d b c
c

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 11: A pentagonal prism of base side 25 mm and axis 50 mm is resting on HP on one of its
rectangular faces, with its axis perpendicular to VP. Draw its projections.

Steps: (i) Draw the front view first, since it shows the true shape of the base of the solid.
(ii) Project the top from the front view.

X Y

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Example Problems - Axis inclined to HP and parallel to VP

Problem 12: A square prism of base side 35 mm and axis length 60 mm is resting on one of its
base edges on the HP with its axis inclined at 30° to the HP and parallel to the VP. Draw its top
and front views.
Steps:(i) Assume the axis is of the solid is kept perpendicular to HP and parallel to VP. Draw
the top view first which has the true shape of the base of the solid and then draw
the front view.
(ii) The front view is tilted and reproduced to the given inclination of axis with HP
(300). Note that the angle of the base edge is 600 (900 – 300). The base edge can be
titled 600 first and the proceeded with further steps also. (This is easier and faster).
(iii) Project the front view to get the top view

FINAL FRONT VIEW

300 600

FINAL TOP VIEW

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 13: A hexagonal prism of base side 30 mm axis length 60 mm is resting on HP on one
of its base sides with its axis inclined at 40° to HP and parallel to VP. Draw its projections.

400 500

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 14: A hexagonal pyramid of base edge 40 mm and altitude 80 mm rests on one of its
base edges on the HP with its axis inclined at 300 to the HP and parallel to the V.P. Draw its top
and front views.

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 15: A pentagonal pyramid side of base 30 mm and axis 45 mm long rests with one of its
corners on HP such that the base is inclined at an angle of 60˚ to HP and one side of base is
perpendicular to VP. Draw its projections.

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Problem 16: A cone of base diameter 40 mm and axis length 80 mm is resting on HP with its axis
inclined at an angle of 30˚ to HP and parallel to VP. Draw its projections.

All Dimensions are in mm

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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Assignment Problem in Projection of Solids

1. A square prism of side 25 mm and axis 50 mm is resting on its base on HP such that, two
base edges are perpendicular to VP. Draw its projections.
2. A hexagonal prism of side 25 mm and axis 70 mm is resting on its base on HP, such that,
two of its base edges are parallel to VP. Draw its projections.
3. A square prism of side 25 mm and axis 60 mm is resting on its base on HP such that, two
base edges are equally inclined to VP. Draw its projections.
4. A pentagonal prism of side 25 mm and axis 65 mm is resting on HP with one of its
rectangular faces, such that one its base edges is parallel to HP. Draw its projections.
5. A triangular prism of side 30 mm and axis length of 60 mm is resting on HP with a face
parallel to VP. Draw the projections.
6. A square prism of base side 30 mm and axis 55 mm is resting on HP on one of its
rectangular faces, with its axis perpendicular to VP. Draw its projections.
7. A triangular prism of base side 30 mm and axis 60 mm is resting on HP on one of its
rectangular faces, with its axis perpendicular to VP. Draw its projections.
8. A hexagonal prism of base side 20 mm and axis 45 mm is resting on HP on one of its
rectangular faces, with its axis perpendicular to VP. Draw its projections.
9. A square prism of base side 30 mm and axis length 60 mm is resting on one of its base
edges on the HP with its axis inclined at 45° to the HP and parallel to the VP. Draw its top
and front views.
10. A hexagonal prism of base side 20 mm axis length 50 mm is resting on HP on one of its
base sides with its axis inclined at 30° to HP and parallel to VP. Draw its projections.
11. A hexagonal pyramid of base edge 20 mm and altitude 50 mm rests on one of its base
edges on the HP with its axis inclined at 300 to the HP and parallel to the V.P. Draw its
top and front views.
12. A pentagonal pyramid side of base 25 mm and axis 650 mm long rests with one of its
corners on HP such that the base is inclined at an angle of 30˚ to HP and one side of base
is perpendicular to VP. Draw its projections.

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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SECTION OF SOLIDS
1. Section Plane inclined to HP and Perpendicular to VP

2. Section Plane inclined to VP and Perpendicular to HP

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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SECTIONAL PROJECTION OF SOLIDS

Frustum: The portion of a cone or pyramid which remains after its upper part has been cut off by
a plane parallel to its base,
Truncated solid: When a solid (prism/cylinder/pyramid/cone) is cut by a cutting plane inclined
to its base (not parallel), the remaining portion obtained after removing the top portion is called
the Truncated Solid.

EXAMPLE PROBLEM 1:
SECTIONAL PROJECTION OF A SQUARE PYRAMID

A square pyramid of side 40 mm and axis length 80 mm lying on HP with a side parallel to VP
and 30 mm in front of it. It is cut by a plane perpendicular to VP and inclined to HP at an angle
450 passing through the midpoint of the axis. Draw the front view, sectional top view and true
shape of the section.

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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EXAMPLE PROBLEM 2: SECTIONAL PROJECTION OF A PRISM

Draw the front view, sectional top view and true shape of the section of a hexagonal prism of side
25 mm and 60 mm high lying on HP with a nearest side parallel to VP and 30 mm in front of it.
It is cut by a plane perpendicular to VP and inclined to HP at an angle 450 passing through a point
on the axis 50 mm above the base.

Solution:

(i) Draw the top view and front view of the hexagonal prism of base 40mm and axis 80mm.
(ii) Mark the point on the axis 50mm above the base and draw a cutting plane line through this
point inclined at an angle 450 to HP.
(iii) Mark the numbering and point of intersection in front view and top view and transfer these
points in the section plan and draw section lines in the cut portion in the top view.
(iv) Project the projection at right angle to the cutting plane line from the interacting points
and complete the true shape. Take distances equal to the corresponding distances on the
plan and join the point and complete the true shape by drawing section lines.

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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EXAMPLE PROBLEM 3: SECTIONAL PROJECTION OF CYLINDER

A cylinder 40 mm diameter and 90 mm long is resting on its base on ground. It is cut by a section
perpendicular to VP. The VP of which cuts the axis at a point 45 mm from the base and makes an
angle 450 to HP. Draw the sectional front view, sectional top view and true shape of the section.
Solution:

(i) Draw the top view and front view of the cylinder with 80 mm diameter and 90 mm long.
(ii) Draw a section plane in such a way that it cuts the axis of the cylinder in front view at a
distance 45 mm base and makes 450 to HP.
(iii) Draw the projection for sectional top view and true sectional view as shown below

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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EXAMPLE PROBLEM 4: A hexagonal pyramid of base side 30 mm and height 65 mm is

resting on its base on the HP with two of its base sides parallel to VP. It is cut by a sectional plane
inclined at 45° to HP, intersecting the axis at a point 25 mm above the base. Draw the front view,
sectional top view and true shape of the section.

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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EXAMPLE PROBLEM 5: A cone of base 75 mm diameter and axis 80 mm long is resting

on its base on the HP. It is cut by a sectional plane inclined at 45° to HP, intersecting the axis at a
point 35 mm from the apex. Draw the front view, sectional top view and true shape of the section.

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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ASSIGNMENT PROBLEMS – SECTION OF SOLIDS

1. A square pyramid of side 40 mm and axis length 80 mm lying on HP with a side parallel
to VP and 30 mm in front of it. It is cut by a plane perpendicular to VP and inclined to HP
at an angle 450 passing through the midpoint of the axis. Draw the front view, sectional
top view and true shape of the section.
2. A hexagonal pyramid of base side 25 mm and height 60 mm rests vertically on its base on
the ground with two of its base sides parallel to VP. It is cut by a sectional plane inclined
at 30° to HP and perpendicular to VP and meeting the axis at the midpoint. Draw the front
view, sectional top view and true shape of the section.
3. A pentagonal pyramid of base 30 mm and axis 80 mm is resting with its base on H.P and
one of the base edges is perpendicular to V.P. The section plane is parallel to H.P and
passing through the axis at a point 50mm above the base. Draw the front view, sectional
top view and true shape of the section.
4. A hexagonal pyramid, edge of base 30 mm and height 70 mm, rests on its base on ground
plane with one of its base edge parallel to VP. A section plane parallel to HP cuts the
pyramid bisecting its axis. Draw its front view and sectional top view.
5. A cylinder 60 mm in diameter and 70 mm long is resting on its base on ground. It is cut
by a section plane perpendicular to VP and cuts the axis at a point 45 mm from the base
making an angle of 400 to HP. Draw the front view, sectional top view and true shape of
the section.

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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PROJECTION OF SOLIDS
PART – A [2 Mark Questions and Answers]
1. Define Polyhedra. Give examples.
A three-dimensional shape with flat polygonal faces, straight edges, and sharp corners or
vertices is called a polyhedron. Common examples are cubes, prisms, pyramids. However,
cylinders, cones and spheres are not polyhedrons since they do not have polygonal faces.
The plural of a polyhedron is called polyhedra.
(i) Cube
(ii) Tetrahedron
(iii) Octahedron
(iv) Prism
(v) Pyramid
2. Define solids of revolution. Give examples.
A solid of revolution is a solid generated by rotating a plane area about its axis.
Following are the solids of revolution.
(i) Cylinder
(ii) Cone
(iii) Sphere
3. Cylinder, cones and spheres are not polyhedra since they do not have polygonal faces.
(True / False)

Ans: True

4. Define a prism. Give examples

A prism is a polyhedron having two equal and similar end faces called top face and bottom
face joined by other faces which are normally rectangles. The imaginary line joining the
centres of the faces is called the axis.
(i) Triangular prism
(ii) Rectangular prism
(iii) Square prism
(iv) Pentagonal prism
(v) Octagonal prism

5. Define a pyramid. Give examples

A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the
apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with
polygonal base.
(i) Triangular pyramid
(i) Rectangular pyramid
(ii) Square pyramid
(iii) Pentagonal pyramid
(iv) Octagonal pyramid

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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6. Give the formula to find the included angle of a polygon. Give an example
180°(n) – 360°
Included angles of a Regular Polygon = -----------------------
n – no of sides
n

(180° x 5) – 360°
Included angles of a pentagon = ------------------------ = 108o
5

7. Give the formula to find the excluded angle (Other side of the included angle) of a
polygon.
Excluded Angle = 180 – Included angle
8. Draw a freehand sketch of the front, right side and top view of a cylinder.

Side
Front View
View

Top
View

9. When the axis of solid is perpendicular to H.P, the ______view should be drawn first
and ____ view is then projected from it.

(i) front, top

(ii) top, side
(iii) side, front
(iv) top, front

Ans: top, front

10. When the axis of solid is perpendicular to V.P, the ______view should be drawn first
and ____ view is then projected from it.

(i) front, top

(ii) top, side
(iii) side, front
(iv) top, front

Ans: front, top

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering

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11. To show the surface of section, hatching lines are drawn at ------------------------
(i) 30°
(ii) 45°
(iii) 60°
(iv) 90°
Ans: (ii) 450
12. 12- A right circular cone is placed on HP on its base. A cutting plane parallel to
horizontal plane cuts the cone, the shape of sectional view is
(i) an ellipse
(ii) a circle
(iii) a parabola
(iv) a hyperbola
Ans: (ii) a circle
13. Define frustrum and truncated solids.
Frustum: The portion of a cone or pyramid which remains after its upper part has been
cut off by a plane parallel to its base,
Truncated solid: When a solid (prism/cylinder/pyramid/cone) is cut by a cutting plane
inclined to its base (not parallel), the remaining portion obtained after removing the top
portion is called the Truncated Solid.

Engineering Graphics - Projection of Solids Dr. M. Ramakrishnan – Professor, Mechanical Engineering