Volume of a solid

The volume of a solid is a measure of the space it takes up. The cube is used as the shape for the basic unit of volume. A cube of edge 1 metre has a volume of 1 cube metre or 1 m³. A cube of edge 1 centimetre has a volume of 1 cubic centimeter  or  1 cm^3.

It is different to measure volume directly. One way is to build a copy of the solid using basic units. For example, to measure the volume of the 6 cm by 3 cm by 4 cm cuboid in the figure below, a copy can be built from 1 cm³ cubes.

Units of volume

The cubic metre, m³, is the basic unit of volume.

1 m = 100 cm

1 m³ = (100 X 100 X 100) cm³ = 1 000 000 cm³

Similarly

1 cm = 10 mm

1 cm³ = (10 X 10 x 10) mm³ = 1 000 mm³

When calculating problems about volume, make sure that all dimension are in the same units.

Volume of Cuboids and Cubes

Cuboid

Notice that

1. the 6 cm by 3 cm by 4 cm cuboid in the above figure has a volume of 72 cm³
2. 6 X 3 X 4 – 72.

We can the volume of any cuboid by finding the product of its length, breadth and height:

volume of cuboid = length X breadth X height

                                = area of base X height

                                = area of end face X length

                                = area of side face X breadth

Cube

Similarly, for a cube of side s,

Volume of cube = (side) X (side) X (side)

= s X s X s

= s³

Example 1

Calculate the volume of a rectangular box which measures 30 cm X 15 cm X 10 cm.

Volume of box = (30 X 15 X 10)cm³

= 4 500 cm³

Example 2

A rectangular room 4 m long by 3 m wide contains 30 m³ of air.

Calculate the height of the room.

Volume of room = 30 m³

area of floor (base) = 4 m X 3 m = 12 m²

height of room = 30/12 m = 2½ m

Capacity of containers

The capacity of a container is the measure of the space inside it. The basic unit of capacity is the litre. 1 litre of water will just fill a 10 cm by 10 cm by 10 cm cubic container.

Therefore in practice,

1 liter = (10 X 10 X 10) cm³ = 1 000 cm³

The table below shows the relation between units of capacity and units of volume.

Example

How many litres of water dos a 5 m X 4 m X 3 m tank hold?

volume of tank = (5 X 4 X 3) m³ = 60 m³

but, 1 m³ = 1 000 litres

capacity of tank = 60 X 1 000 litres

= 60 000 litres