The document discusses the development of surfaces, which is the process of laying out the entire surface of a 3D object onto a 2D plane. It describes various methods for developing different types of surfaces and solids, including parallel line development for prisms and cylinders, radial line development for cones and pyramids, and triangulation for more complex shapes. It then provides examples of developing specific objects like prisms, cylinders, pyramids, and cones.

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DEVELOPMENT OF SURFACES
Introduction
A layout of the complete surface of a three dimensional object on a plane is called the
development of the surface or flat pattern of the object. The development of surfaces
is very important in the fabrication of articles made of sheet metal.
The objects such as containers, boxes, boilers, hoppers, vessels, funnels, trays etc.,
are made of sheet metal by using the principle of development of surfaces.
In making the development of a surface, an opening of the surface should be
determined first.
Every line used in making the development must represent the true length of the line
(edge) on the object.
“The development of surface of an object means the unrolling and unfolding of all
surfaces of the object on a plane.”
“If the surface of a solid is laid out on a plain surface, the shape thus obtained is
called the development of that solid.”
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.
Importance of Development:
Knowledge of development is very useful in sheet metal work, construction of storage
vessels, chemical vessels, boilers, and chimneys. Such vessels are manufactured
from plates that are cut according to these developments and then properly bend into
desired shaped. The joints are then welded or riveted.
Principle of Development:
Every line on the development should show the true length of the corresponding line
on the surface which is developed.
Objective in this topic:
To learn methods of development of surfaces of different solids, their sections and
frustums.
Methods of Development
The method to be followed for making the development of a solid depends upon the
nature of its lateral surfaces. Based on the classification of solids, the following are the
methods of development.

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1. Parallel-line Development
It is used for developing prisms and single curved surfaces like cylinders/prism in
which all the edges / generators of lateral surfaces are parallel to each other.
2. Radial-line Development
It is employed for pyramids and single curved surfaces like cones in which the apex is
taken as centre and the slant edge or generator (which are the true lengths)as radius
for its development.
3. Triangulation method:
This is generally used for polyhedron, single curved surfaces, and warped surfaces.
4. Approximate development:
In this, the shapes obtained are only approximate. After joining, the part is stretched
or distorted to obtain the final shape
Parallel-line developments are made from common solids that are composed of
parallel lateral edges or elements. e.g. Prisms and cylinders
The cylinder is positioned such that one element lies on the development plane. The
cylinder is then unrolled until it is flat on the development plane. The base and top of
the cylinder are circles, with a circumference equal to the length of the development.
All elements of the cylinder are parallel and are perpendicular to the base and the top.

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When cylinders are developed, all elements are parallel and any perpendicular section
appears as a stretch-out line that is perpendicular to the elements.
Developments of objects with parallel elements or parallel lateral edges begins by
constructing a stretch-out line that is parallel to a right section of the object and is
therefore, perpendicular to the elements or lateral edges.
Radial-line developments are made from figures such as cones and pyramids. In
the development, all the elements of the figure become radial lines that have the
vertex as their origin.
The cone is positioned such that one element lies on the development plane. The
cone is then unrolled until it is flat on the development plane. One end of all the
elements is at the vertex of the cone. The other ends describe a curved line.

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The base of the cone is a circle, with a circumference equal to the length of the curved
line.
Triangulation developments are made from polyhedrons, single curved surfaces,
and wrapped surfaces. The development involve subdividing any ruled surface into
a series of triangular areas. If each side of every triangle is true length, any number of
triangles can be connected into a flat plane to form a development Triangulation for
single curved surfaces increases in accuracy through the use of smaller and more
numerous triangles.
Triangulation developments of wrapped surfaces produces only approximate of those
surfaces.
Approximate developments are used for double curved surfaces, such as spheres.
Approximate developments are constructed through the use of conical sections of the
object.Approximate developments the material of the object is then stretched through
various machine applications to produce the development of the object.

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DEVELOPMENT OF SURFACES – PRISM PROBLEMS
1. A Square prism of base side 40 mm and axis length 50 mm is resting on HP
on one of its base with a side of base inclined at 350 to VP. It is cut by a plane
inclined at 300 to HP and perpendicular to VP and is bisecting the axis. Draw the
development of the remaining portion of the prism.

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2. A pentagonal prism of base side 25 mm and height 55 mm is cut by a plane
perpendicular to VP and 300 to HP and passing through the axis 30 mm above
the base, draw the lateral surfaces development in the lower portion of the solid.
3. A pentagonal prism of base side 25 mm and height 60 mm is resting on the
ground with one of its base edge parallel to VP .Find graphically the shortest
distance of the string which connect one end of the lateral edge with the other
end of the same edge, covering all the lateral surfaces of the solid. Also trace
the points on the development.

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4. A hexagonal prism of base side 30 mm and height 60 mm is resting on the
ground with one of its vertical faces perpendicular to VP .it is cut by a plane
inclined at 500 to HP and perpendicular to VP and meets the axis of the prism at
a distance of 10 mm from the top end. Draw the development of the lateral
surfaces
DEVELOPMENT OF SURFACES – CYLINDER
5. Draw the development of the lateral surfaces of the lower portion of a cylinder
of diameter 45 mm and height 60 mm when sectioned by a plane inclined at
400to HP and perpendicular to VP and bisecting the axis.

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6. A cylinder of diameter 50 mm and axis height 65 mm is cut by a plane inclined
at 600 to the HP and bisecting the axis. Draw the development of the lateral
surfaces
7. A cylinder of diameter 40 mm and axis height 75 mm is cut by a plane
perpendicular to VP inclined at 550 to HP meeting the axis at the top face. Draw
the development of the lateral surfaces of solid.

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DEVELOPMENT OF SURFACES – PYRAMIDS
8. A pentagonal pyramid of base 25 mm side and height 65 mm stands with its
base on the HP such that one of its base edges is parallel to the VP. It is cut by a
section plane perpendicular to the VP and inclined at 300 to the HP, bisecting
the axis. Draw the development of the lateral surfaces of solid.
9.A pentagonal pyramid of base 25 mm side and height 60 mm lying on the HP
on its base such that one of its base edges is parallel to and far away from the
VP. It is cut by a section plane one is perpendicular to the VP and inclined at 400
to the HP, and meeting the axis at 14 mm from the base the other plane is
parallel to HP and perpendicular to VP meeting the axis distance of 28 mm from
the base. Draw the development of the lateral surfaces of solid.

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10. A square pyramid of base side 30 mm and altitude 65 mm is resting on HP
on its base with a side of the base inclined at 250 to VP. It is cut by a plane
inclined 350 to HP and perpendicular to VP and bisects the axis. Draw the
development of the remaining surfaces of solid
11. A hexagonal pyramid of base 25 mm side and height 50 mm rests on its base
with one base edge parallel to VP.A string is wound around the surfaces of the
pyramid starting from the left extreme point of the base and ending at the same.
Find the shortest length of the string required. Also trace the path of the string
in the projection.

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12. A hexagonal pyramid of base side 25 mm and altitude 60 mm is resting
vertically on its base on the ground with two of the of the sides of the base
perpendicular to the VP. It is cut by a plane perpendicular to the VP and inclined
at 450 to the HP. The plane bisects the axis of the pyramid. Draw the
development of the lateral surfaces of solid.
13.A square pyramid of base side 30 mm and height 50 mm rests on its base on
the HP, with a base edge parallel to VP. It is cut by a plane perpendicular to VP
and inclined 500to HP meeting the axis 30 mm above HP. Draw the development
of the lateral surfaces.

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14. A pentagonal pyramid of base 30 mm side and height 60 mm stands with its
base on the HP on its base edges is perpendicular to the VP. It is cut by a
section plane perpendicular to the VP and parallel to the HP, and meets the axis
at a distance of 25 mm from the vertex. Draw the development of the lateral
surfaces of solid.
15. A hexagonal pyramid of side of base 30mm and altitude 75 mm rests on its
base on HP, such that a base edge is parallel to VP.it is cut by twocutting planes
perpendicular to VP. One of the planes is inclined at 300 to HP and meeting the
axis at a point 40 mm from the base.The other plane is curved of 30 mm radius
with the right corner of the base as centre. Draw the development of the lateral
surfaces.

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16. A square pyramid of base side 35 mm and height 70 mm rests on its base on
the HP, such that two adjacent sides of the base are equally inclined to VP. It is
sectioned by a plane perpendicular to VP, inclined 300to HP and passing
through the midpoint of the axis. Draw the development of the lateral surfaces.
17. Draw the development of the lateral surfaces of a hexagonal pyramid with a
40 mm base side and a 60 mm long axis ,which is resting on the base in the HP
such that an edge of the base is perpendicular to VP when an auxiliary inclined
plane whose VT makes on angle 600 HP and bisecting the axis.

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DEVELOPMENT OF SURFACES – CONE
18. A cone of base side 60mm and height 70 mm rests on its base on the HP, It
is sectioned by a plane perpendicular both HP and VP, and 10 mm away from
the axis. Draw the development of the lateral surfaces.
19. A cone of base side 50 mm and height 65 mm rests on its base on the HP, It
is sectioned by a plane perpendicular VP and inclined at 300 to HP bisect the
axis of the cone. Draw the development of the lateral surface.

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20.A right circular cone of diameter 50 mm axis height 60 mm is string on the
ground with its base. Calculate the shortest length of a string required to wound
round the lateral surface of the solid starting from one extreme point and ending
at the same point. Also trace the points on to the projections.
21. A cone of base side 50 mm and height 75 mm rests on its base on the HP, It
is sectioned by a plane perpendicular VP and parallel to HP at a distance 20 mm
from the vertex. It is also cut by a plane inclined at 400to the base and meeting
the axis at a point 22 mm above the base. Draw the development of the lateral
surface.

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22.A cone of base side 45 mm and height 70 mm rests on its base on the HP, It is
sectioned by a plane perpendicular VP 300to HP and bisecting the axis. Draw the
development of the lateral surfaces
23. A cone of base side 50 mm and height 60 mm rests on its base on the HP, It
is sectioned by a plane perpendicular VP ,parallel to one of the generators and
passing through a point on the axis at a distance of 22 mm from the apex. Draw
the development of the lateral surfaces

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24. Draw the development of the tray whose pictorial view as shown in fig
25. Draw the development of a duct as shown.

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26. Draw the development of the three pipes forming a Y shape as shown. All the pipes
are diameters of 40 mm. The max. Height of the vertical pipe is 50 mm. The angle
between the axes of the inclined pipes is 800.
27. An offset fitting is made up of three pipes of diameter 40 mm each. The total length
of the fitting is 90 mm and the offset is 55 mm. Draw the lateral surface of the pipes.

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28. An elbow is made up of three pipes are diameter 40 mm each fitted as shown.
The shorter arm of both the vertical and horizontal pipes has the same length of
20 mm. Draw the development of pipes forming the elbow.
29. A funnel is made up of a truncated cone and a cut cylinder as shown. The
cone is of base diameter 60 mm and altitude 70 mm. They are fitted as shown.
Draw the development of funnel forming cone.