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05 Volume of SolidsBy studying this lesson you will acquire knowledge on thefollowing :

Let us recall your knowledge on the volumes of cubes, cuboids,prisms and cylinders.

(i) Cube

(iii) prism (iv) Cylinder

(ii) Cuboid

a a

b

a

a

lh

a

h

r

The volume of a solid with a uniform cross section can be found bytaking 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

c

Calculating the volume of a square based right pyramid

Calculating the volume of a coneCalculating the volume of a sphere

(using the formula)

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Cube : Area of cross section × height

Cuboid : Area of cross section × height

Triangular Prism: Area of cross section × length

Cylinder : Area of cross section × height

Volume of a solid with uniform cross section = Area of cross section × length

Make a hollow cube of length 12 cm a side and a hollow right pyramid with asquare base of 12 cm a side and height 12 cm.

5 .1 Volume of a square based pyramid

Activity 5.1

12 cm

12 cm

Cube Pyramid

=

==

==

=

=

=

=

=

(a × a) × a

(a × b) × c

a × aa

×c

1 12 2

a

� �r r2 2

ah

h h

h×l

×

l

ababc

Cubic units

Cubic units

Cubic units

Cubic units

Cubic units

Cubic units

Cubic units

Cubic units

2

3

●

●

●

●

●

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V olume of the Pyramid × 3 =

V olume of the Pyramid =

Example 1

12 cm12 cm

12 cm

12 cm

=

= 12 cm 12 cm 12 cm

12 cm

V olume of the Pyramid =

V olume of the Pyramid =

V olume of the cube

v olume of the cube

area of the base × perpendicular height

area of the base × height

area of base × perpendicular height

×

×

×

×

576 cm

×

3

× ×

=

1313

13

1313

12cm

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.as follows.

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The volume of the square based pyramid

Example 3

area of the base × perpendicular height

area of the base × 9 cm

Area of the base

Length of a side of the base

The length of a side of a square based pyramid is 12 cm. The volume of the pyramid is 384 cm . Find the perpendicular height of the pyramid. Find theperpendicular height of a triangular face of the pyramid.

=

=

=

=

=

==

588 cm

588 cm

588 cm

9 cm588 × 3 cm

196 cm

196 cm 14 cm

3

3

3

3

3

12 cm6 cm

h x

2

×

×

13

13

Example 2

The volume of a square based pyramid is 588 cm3, It’s perpendicularheight is 9 cm. Find the length of a side of the base.

9 cm

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Find the volume of a pyramid with a square base of 6 cm a side and of perpendicular height 4 cm.

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.

(1)

(2)

(3)

(4)

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.

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.

According to the diagram, if the perpendicular height ofa triangular face is cm,by applying Pythagors’ theorem,

x

Height of a triangular face = 10 cm

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5.2 Cone

(5)

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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5.3 Sphere

First find a small sphere and measure its’diameter. Make acylinder having its height and the radius of the cross sectionequal to the diameter and the radius of the sphererespectively.Put the sphere carefully into the cylinder as shown in thefigure.

Activity

Find the volume of a cone of base radius 7 cm and perpendicular height 10 cm.

Find the volume of a cone of base radius 7 cm and perpendicular height 15 cm

Find the radius of the base of a cone of volume 616 cm3 and perpendicular height 12 cm.

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.

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.

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.

2

(1)

(2)

(3)

(4)

(5)

(6)

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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.

Volume of the circumscribed cylinder - Volume of the shaded portion.

The volume of the sphere =

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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