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Unit C4 Section 1
Volumes of Cubes, Cuboids, Cylinders and Prisms

The volume of a cube is given by

V = a3

where a is the length of each side of the cube.

For a cuboid the volume is given by

V = abc

where a, b and c are the lengths shown in the diagram.

The volume of a cylinder is given by

V = πr2h

where r is the radius of the cylinder and h is its height.

The volume of a triangular prism can be expressed in two ways,
as

V = Al

where A is the area of the end and l the length of the prism,
or as

V = bhl

where b is the base of the triangle and h is the height of the triangle.

Worked Examples

1

The diagram shows a truck.

Find the volume of the load-carrying part of the truck.

The load-carrying part of the truck is represented by a cuboid, so its volume is given by

V = 2 × 2.5 × 4
= 20 m3
2

The cylindrical body of a fire extinguisher has the dimensions shown in the diagram. Find the maximum volume of water the extinguisher could hold.

The body of the extinguisher is a cylinder with radius 10 cm and height 60 cm, so its volume is given by

V = π × 102 × 60
= 18 850 cm3 (to the nearest cm3)
3

A traffic calming road hump (sleeping policeman) is made of concrete and has the dimensions shown in the diagram. Find the volume of concrete needed to make one road hump.

The shape is a triangular prism with b = 80, h = 10 and l = 300 cm. So its volume is given by

V = × 80 × 10 × 300
= 120 000 cm3
4

The diagram below, not drawn to scale, shows a container in the shape of a rectangular prism.

The base of the container has a length of 75 cm and a width of 40 cm.

(a)

Calculate the area, in cm2 , of the base of the container.

Area of base = 75 × 40 = 3000 cm2

Water is poured into the container, reaching a height of 15 cm.
(b)

Calculate, in cm3, the volume of water in the container.

Volume = 15 × 3000
= 45 000 cm3
(c)

If the container holds 84 litres when full, calculate the height, h, in cm, of the water when the container is full.

When full, the tank holds 84 × 1000 cm3 of water, so

h × 3000 = 84 000
h =
= 28 cm

Exercises

Find the volume of each solid shown below.

(a)
cm³
(b)
cm³
(c)
m³
(d)
mm³ (to the nearest mm³)
(e)
m³ (to the nearest m³)
(f)
cm³ (to the nearest cm³)
(g)
cm³
(h)
cm³
(i)
m³
(a)

Find the volume of the litter bin shown in the diagram, in m3 to 2 decimal places.

m3
(b)

Find the volume of rubbish that can be put in the bin, if it must all be below the level of the hole in the side, in m3 to 2 decimal places.

m3

A concrete pillar is a cylinder with a radius of 20 cm and a height of 2 m.

Give your answers rounded to the nearest cm3

(a)

Find the volume of the pillar.

cm3
The pillar is made of concrete, but contains 10 steel rods of length 1.8 m and diameter 1.2 cm.
(b)

Find the volume of one of the rods and the volume of steel in the pillar.

volume of one of the rods = cm3

volume of steel in the pillar = cm3

(c)

Find the volume of concrete contained in the pillar.

cm3

Find the volume of each prism below.

(a)
cm³
(b)
cm³
(c)
cm³
(d)
cm³

The diagram shows the cross section of a skip that is 15 m in length and is used to deliver sand to building sites. Find the volume of sand in the skip when it is filled level to the top.

m3

Tomato soup is sold in cylindrical tins.

Each tin has a base radius of 3.5 cm and a height of 12 cm.

(a)

Calculate the volume of soup in a full tin.
Take π to be 3.14 or use the π key on your calculator.

cm3 (to the nearest cm3)
(b)

Bradley has a full tin of tomato soup for dinner. He pours the soup into a cylindrical bowl of radius 7 cm.

What is the depth of the soup in the bowl?

cm (to the nearest cm)

The diagram represents a swimming pool.
The pool has vertical sides.
The pool is 8 m wide.

(a)

Calculate the area of the shaded cross section.

m2

The swimming pool is completely filled with water.
(b)

Calculate the volume of water in the pool.

m3

64 m3 leaks out of the pool.
(c)

Calculate the distance by which the water level falls.

m

The diagram represents a carton in the shape of a cuboid.

(a)

Calculate the volume of the carton.

cm3

There are 125 grams of sweets in a full carton.
John has to design a new carton that will contain 100 grams of sweets when it is full.
(b)
(i)

Work out the volume of the new carton.

cm3
(ii)

Express the weight of the new carton as a percentage of the weight of the carton shown.

%

The new carton is in the shape of a cuboid.
The base of the new carton measures 7 cm by 6 cm.
(c)
(i)

Work out the area of the base of the new carton.

cm2
(ii)

Calculate the height of the new carton.

cm (to the nearest mm)