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1
3. DEVELOPMENT OF SURFACES OF SOLIDS
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ENGINEERING APPLICATIONS OF THE PRINCIPLES OF
PROJECTIONS OF SOLIDES.

1. SECTIONS OF SOLIDS.
2. DEVELOPMENT.
3. INTERSECTIONS.

STUDY CAREFULLY THE ILLUSTRATIONS GIVEN ON NEXT
SIX PAGES !
3
SECTIONING A SOLID. An object ( here a solid ) is
cut by some imaginary cutting plane to
understand internal details of that object.
The action of cutting is called SECTIONING a
solid The plane of cutting is called SECTION
PLANE.
Two cutting actions means section planes are
recommended. A) Section Plane perpendicular to
Vp and inclined to Hp. ( This is a
definition of an Aux. Inclined Plane i.e.
A.I.P.) NOTE- This section plane appears
as a straight line in FV. B)
Section Plane perpendicular to Hp and inclined to
Vp. ( This is a definition of an Aux.
Vertical Plane i.e. A.V.P.) NOTE- This
section plane appears as a
straight line in TV. Remember- 1. After
launching a section plane either in FV or
TV, the part towards observer is assumed to
be removed. 2. As far as possible the smaller
part is assumed to be removed.
(A)
(B)
4
For TV
ILLUSTRATION SHOWING IMPORTANT TERMS IN
SECTIONING.
For True Shape
SECTION PLANE
TRUE SHAPE Of SECTION
x
y
Apparent Shape of section
SECTION LINES (450 to XY)
SECTIONAL T.V.
5
Typical Section Planes Typical Shapes Of
Sections.
Section Plane Through Generators
Ellipse
Triangle
Section Plane Through Apex
Parabola
Section Plane Parallel to Axis.
Section Plane Parallel to end generator.
Hyperbola
Ellipse
Trapezium
Cylinder through generators.
Sq. Pyramid through all slant edges
6
DEVELOPMENT OF SURFACES OF SOLIDS. MEANING- ASSU
ME OBJECT HOLLOW AND MADE-UP OF THIN SHEET. CUT
OPEN IT FROM ONE SIDE AND UNFOLD THE SHEET
COMPLETELY. THEN THE SHAPE OF THAT UNFOLDED SHEET
IS CALLED DEVELOPMENT OF LATERLAL SUEFACES OF
THAT OBJECT OR SOLID.
LATERLAL SURFACE IS THE SURFACE EXCLUDING
SOLIDS TOP BASE.
ENGINEERING APLICATION THERE ARE SO MANY
PRODUCTS OR OBJECTS WHICH ARE DIFFICULT TO
MANUFACTURE BY CONVENTIONAL MANUFACTURING
PROCESSES, BECAUSE OF THEIR SHAPES AND SIZES.
THOSE ARE FABRICATED IN SHEET METAL INDUSTRY BY
USING DEVELOPMENT TECHNIQUE. THERE IS A VAST
RANGE OF SUCH OBJECTS. EXAMPLES- Boiler Shells
chimneys, Pressure Vessels, Shovels, Trays,
Boxes Cartons, Feeding Hoppers, Large Pipe
sections, Body Parts of automotives, Ships,
Aeroplanes and many more.
WHAT IS OUR OBJECTIVE IN THIS TOPIC ?
To learn methods of development of surfaces
of different solids, their sections and frustums.
1. Development is different drawing than
PROJECTIONS. 2. It is a shape showing AREA, means
its a 2-D plain drawing. 3. Hence all dimensions
of it must be TRUE dimensions. 4. As it is
representing shape of an un-folded sheet, no
edges can remain hidden And hence DOTTED
LINES are never shown on development.
But before going ahead, note following Important
points.
Study illustrations given on next page carefully.
7
Development of lateral surfaces of different
solids. (Lateral surface is the surface excluding
top base)
Cylinder A Rectangle
Pyramids (No.of triangles)
Cone (Sector of circle)
L
?
H Height D base diameter
Prisms No.of Rectangles
L Slant edge. S Edge of base
H Height S Edge of base
Cube Six Squares.
Tetrahedron Four Equilateral Triangles
All sides equal in length
8
DEVELOPMENT OF FRUSTUM OF CONE
DEVELOPMENT OF FRUSTUM OF SQUARE PYRAMID
Base side
Top side
?
R Base circle radius of cone L Slant height of
cone L1 Slant height of cut part.
L Slant edge of pyramid L1 Slant edge of cut
part.
9
TRUE SHAPE
a b e c d
X
Y
DEVELOPMENT
10
TRUE SHAPE OF SECTION
A
SECTIONAL S.V
B
o
SECTION PLANE
DEVELOPMENT
C
D
E
X
Y
g hf ae bd c
F
G
H
A
SECTIONAL T.V
11
Problem 3 A cone 40mm diameter and 50 mm axis is
resting on one generator on Hp( lying on Hp)
which is // to Vp.. Draw its projections.It is
cut by a horizontal section plane through its
base center. Draw sectional TV, development of
the surface of the remaining part of cone.
Follow similar solution steps for Sec.views -
True shape Development as per previous problem!
DEVELOPMENT
o
HORIZONTAL SECTION PLANE
Y
X
o
O
SECTIONAL T.V
(SHOWING TRUE SHAPE OF SECTION)
12
SECTIONAL F.V.
A.V.P300 inclined to Vp Through mid-point of axis.
1,2 3,8
4,7 5,6
AS SECTION PLANE IS IN T.V., CUT OPEN FROM
BOUNDRY EDGE C1 FOR DEVELOPMENT.
TRUE SHAPE OF SECTION
DEVELOPMENT
13
TRUE SHAPE
O
DEVELOPMENT
F.V.
SECTIONAL TOP VIEW.
14
o
15
o
16
(No Transcript)
17
o
1 2 3 4
5 6 7
18
(No Transcript)
19
o
DEVELOPMENT
X
Y

20
Q 15.26 draw the projections of a cone resting
on the ground on its base and show on them, the
shortest path by which a point P, starting from a
point on the circumference of the base and moving
around the cone will return to the same point.
Base ofn cone 65 mm diameter axis 75 mm long.
1
12
11
10
9
8
?103º
7
6
5
4
3
Y
2
2 12
3 11
4 10
5 9
6 8
X
1
7
21
Q e A right circular cone base 30 mm side and
height 50 mm rests on its base on H.P. It is cut
by a section plane perpendicular to the V.P.,
inclined at 45º to the H.P. and bisecting the
axis. Draw the projections of the truncated cone
and develop its lateral surface.
Y
2 12
3 11
4 10
5 9
6 8
X
1
7
22
Q 14.11 A square pyramid, base 40 mm side and
axis 65 mm long, has its base on the HP and all
the edges of the base equally inclined to the VP.
It is cut by a section plane, perpendicular to
the VP, inclined at 45º to the HP and bisecting
the axis. Draw its sectional top view, sectional
side view and true shape of the section. Also
draw its development.
21
31
o
1
41
3
11
4
2
4
O
3
1
b
d
Y
X
c
a
45º
2
d
4
1
1
a
c
o
3
2
b
23
Q 14.14 A pentagonal pyramid , base 30mm side
and axis 60 mm long is lying on one of its
triangular faces on the HP with the axis parallel
to the VP. A vertical section plane, whose HT
bisects the top view of the axis and makes an
angle of 30º with the reference line, cuts the
pyramid removing its top part. Draw the top view,
sectional front view and true shape of the
section and development of the surface of the
remaining portion of the pyramid.
C
o
B
D
5
A
6
1
4
E
3
2
1
6
5
2
Y
X
4
3
A
O
1
2
3
1
31
6
4
6
21
41
5
5
11
51
61
24
Q 14.11 A square pyramid, base 40 mm side and
axis 65 mm long, has its base on the HP with two
edges of the base perpendicular to the VP. It is
cut by a section plane, perpendicular to the VP,
inclined at 45º to the HP and bisecting the axis.
Draw its sectional top view and true shape of the
section. Also draw its development.
3
o
2
1
True length of slant edge
2
2 3
2 3
4
1
O
True length of slant edge
3
1 4
1 4
2
Y
X
a d
b c
a
b
1
2
1
o
3
4
c
d
25
Q.15.11 A right circular cylinder, base 50 mm
diameter and axis 60 mm long, is standing on HP
on its base. It has a square hole of size 25 in
it. The axis of the hole bisects the axis of the
cylinder and is perpendicular to the VP. The
faces of the square hole are equally inclined
with the HP. Draw its projections and develop
lateral surface of the cylinder.
4 10
2 12
3 11
5 9
6 8
7
1
B
B
A
A
C
C
D
D
X
Y
a
c
1
3
5
9
2
4
6
7
8
10
11
12
1
b d
a
c
c
a
b d
26
Q.15.21 A frustum of square pyramid has its base
50 mm side, top 25 mm side and axis 75 mm. Draw
the development of its lateral surface. Also draw
the projections of the frustum (when its axis is
vertical and a side of its base is parallel to
the VP), showing the line joining the mid point
of a top edge of one face with the mid point of
the bottom edge of the opposite face, by the
shortest distance.
o
A
A1
D
True length of slant edge
P
C
R
D1
B
ad
bc
p
A
r
S
C1
s
75
B1
Q
q
a1d1
b1c1
X
Y
A1
d1
c1
p
r
c
d
50
o
25
a
b
s
a1
b1
q
27
THANKS..
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