05 Volume of Solids
By studying this lesson you will acquire knowledge on the
following :
Calculating the volume of a square based right pyramid
(using the formula)
Calculating the volume of a cone
Calculating the volume of a sphere
Let us recall your knowledge on the volumes of cubes, cuboids,
prisms and cylinders.
(i)
Cube
(ii)
Cuboid
b
a
c
a
(iii)
prism
a
a
(iv)
Cylinder
h
h
l
r
a
The volume of a solid with a uniform cross section can be found by
taking the product of the area of the cross section and the length.
The solids given above are all with uniform cross sections
Therefore the volume = cross sectional area × length
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● Cube
: Area of cross section × height
= (a × a) × a Cubic units
=a ×a
2
Cubic units
3
Cubic units
=a
● Cuboid : Area of cross section × height
= (a × b) × c Cubic units
Cubic units
= ab ×c
Cubic units
= abc
1
1
ahl Cubic units
2
2
2
= � r ×h = � r 2 h Cubic units
● Triangular Prism: Area of cross section × length = ah ×l =
● Cylinder : Area of cross section × height
Volume of a solid with uniform cross section = Area of cross section × length
5 .1 Volume of a square based pyramid
Activity 5.1
● Make a hollow cube of length 12 cm a side and a hollow right pyramid with a
square base of 12 cm a side and height 12 cm.
12 cm
12 cm
Pyramid
Cube
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Fill the pyramid completely with soft sand and then put this amount of sand in
to the cube. Repeat this action until the cube is filled with sand. You will
see that this has to be repeated thrice. Therefore the result can be written
as follows.
V olume of the Pyramid × 3 = V olume of the cube
1
× v olume of the cube
3
1
= × area of the base × perpendicular height
3
V olume of the Pyramid =
V olume of the Pyramid =
1
× area of the base × height
3
Example 1
12 cm
12 cm
12 cm
12 cm
12
cm
12 cm
1
V olume of the Pyramid = × area of base × perpendicular height
3
1
= ×12 cm ×12 cm ×12 cm
3
= 576 cm 3
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Example 2
The volume of a square based pyramid is 588 cm3, It’s perpendicular
height is 9 cm. Find the length of a side of the base.
9 cm
The volume of the square based pyramid
= 588 cm3
1
3
× area of the base × perpendicular height = 588 cm
3
1
= 588 cm3
× area of the base × 9 cm
3
588 × 3 cm3
Area of the base
=
9 cm
= 196 cm 2
Length of a side of the base
= 196 cm
= 14 cm
Example 3
The length of a side of a square based pyramid is 12 cm. The volume of the
pyramid is 384 cm3 . Find the perpendicular height of the pyramid. Find the
perpendicular height of a triangular face of the pyramid.
h
x
6 cm
12 cm
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According to the diagram, if the perpendicular height of
a triangular face is x cm,
by applying Pythagors’ theorem,
Height of a triangular face = 10 cm
(1) Find the volume of a pyramid with a square base of
6 cm a side and of perpendicular height 4 cm.
(2) The figure shows a model of a monument. It takes
the shape of a pyramid. Its perpendicular height is 8
cm and a side of its square base is 12 cm. Find the
volume of the model.
(3) A square based pyramid is made of glass and a side of its base is 10 cm. If the
height of the pyramid is 12 cm, find the volume of glass contained in the
pyramid.
(4) The length of a side of the base of a square based pyramid is 8 3 cm. Find
the volume of the pyramid if its perpendicular height is 9 cm.
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(5)
5.2 Cone
Make a container without a lid in the shape of a cone of desired base
radius, with a thick paper. Make a cylinder with the base having the same radius,
and height the saume of a pme as the cone. Fill the cone completely with soft
sand and put that sand into the cylinder. Repeat this until the cylinder is filled
completely. Find how many times this has to be repeated to fill the cylinder com-
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Using Pythagoras’ theorem,
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(1) Find the volume of a cone of base radius 7 cm and perpendicular height
10 cm.
(2) Find the volume of a cone of base radius 7 cm and perpendicular height
2
15 cm
(3) Find the radius of the base of a cone of volume 616 cm3 and perpendicular
height 12 cm.
(4) If the circumfeence of a solid cone is 66 cm, find the radiu s of the base.
If the perpendicular height of the cone is 9 cm find the volume of the cone.
(5) The circumference of the base of a cone is 44 cm. Its slant height is measured
as 58 cm. Find the volume of the cone.
(6) It is necessary to fill sand into a hollow cone of base radius 21 cm and
slant height 35 cm. Find the volume of sand needed to fill the cone.
5.3 Sphere
Activity
First find a small sphere and measure its’diameter. Make a
cylinder having its height and the radius of the cross section
equal to the diameter and the radius of the sphere
respectively.
Put the sphere carefully into the cylinder as shown in the
figure.
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You can see that the sphere does not take up the whole space in the
circumscribed cylinder. The volume of the sphere can be written as follows.
The volume of the sphere = Volume of the circumscribed
cylinder - Volume of the
shaded portion.
To find the volume of the shaded portion, let us make
a hollow cone of the same radius and height as the
cylinder.
Fill the cone completly with soft sand and then put it in
to the cylinder. When the upper part is filled completely,
turn over the cylinder carefully and fill the other side
with the rest of the sand again.
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The radius of the sphere
The volume of the sphere
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7 cm
Find the volume of a metal hemisphere of radius 14 cm. A volume of 1 cm3
of the meterial used to make the hemisphere weighs 5 g. Find the mass of
the hemisphere.
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