Volume of Cuboids and Prisms: Edexcel GCSE Foundation maths revision notes

Volume in the exam

In the exam you are often asked to work out the volume of a cuboid from a diagram, for 2 or 3 marks. One of the marks can be for giving the right units, such as \text{cm}^3

Other questions give a prism with a triangle or a trapezium at each end, or give you the volume and ask for a missing length. These are usually worth 3 marks.

Later in the paper, volume is part of a longer problem for 4 or 5 marks: how many litres a tank holds, how many cups a container fills, or how many small boxes pack into a big one.

Since 2022 a formulae sheet has been given, with the volume of a prism and the area of a trapezium, but learn both formulae anyway. You need to know the area of a rectangle and of a triangle yourself.

What volume measures

Volume is the amount of space inside a solid shape. It is measured by counting cubes. A cube with edges 1 cm long has a volume of 1 cubic centimetre, written 1\text{ cm}^3

Length is measured in units such as cm or m.
Area is measured in square units such as \text{cm}^2 or \text{m}^2
Volume is measured in cubic units such as \text{cm}^3 or \text{m}^3

A cuboid is a box shape whose six faces are all rectangles. To find its volume, multiply its three lengths:

V = l \times w \times h

where l is the length, w is the width and h is the height.

Worked example 1: The volume of a cuboid

A cuboid is 7 cm long, 4 cm wide and 3 cm high. Work out the volume of the cuboid.

The cuboid can be filled with centimetre cubes. The top layer is 7 cubes long and 4 cubes wide.
One layer holds 7 \times 4 = 28 cubes.
Your turn. There are 3 layers. Work out 28 \times 3

Your answer

28 \times 3 = 84, so there are 84 cubes.
This is the same as length \times width \times height: 7 \times 4 \times 3 = 84
Answer: 84\text{ cm}^3

Cubes

A cube is a cuboid whose length, width and height are all the same. A cube with edges 5 cm long has volume 5 \times 5 \times 5 = 125\text{ cm}^3

5 \times 5 \times 5 is written 5^3 and read as "five cubed". Cubing a number means multiplying three copies of it together. It does not mean multiplying by 3

To go back from the volume to the edge, find the number that cubes to give it. A cube with volume 216\text{ cm}^3 has edges 6 cm long, because 6 \times 6 \times 6 = 216

Worked example 2: Finding a missing height

A juice carton is a cuboid with a square base. Each side of the base is 6 cm long. The volume of the carton is 252\text{ cm}^3

Work out the height of the carton.

The base is a square, so the length and the width are both 6 cm. Volume = length \times width \times height, so 6 \times 6 \times h = 252
The area of the base is 6 \times 6 = 36\text{ cm}^2, so 36 \times h = 252
Your turn. Divide to undo the multiplying. Work out 252 \div 36

Your answer

252 \div 36 = 7, so h = 7
Check: 6 \times 6 \times 7 = 252
Answer: 7 cm

Prisms and cross-sections

A prism is a solid that has the same shape all the way through. If you cut straight across it anywhere, you get the same flat shape. That shape is the cross-section.

A cuboid is a prism with a rectangle as its cross-section. A triangular prism has a triangle as its cross-section, like a slice of cheese or a tent.

Think of the prism as layers of the cross-section stacked along its length. So:

V = A \times l

where A is the area of the cross-section and l is the length of the prism.

Rectangle: length \times width.
Triangle: \frac{1}{2} \times base \times height.
Trapezium: \frac{1}{2}(a + b)h, where a and b are the parallel sides.
An L shape: split it into two rectangles and add their areas.

Always find the area of the cross-section first, then multiply by the length. Do not just multiply every number on the diagram.

Worked example 3: A triangular prism

Here is a triangular prism. The cross-section is a right-angled triangle with base 6 cm and height 4 cm. The prism is 9 cm long.

Work out the volume of the prism.

The shaded triangle is the cross-section. It is the same all the way along the 9 cm length.
Your turn. Work out the area of the triangle, \frac{1}{2} \times 6 \times 4

Your answer

\frac{1}{2} \times 6 \times 4 = 12\text{ cm}^2
Multiply by the length: 12 \times 9 = 108
The halving was done once, for the triangle. Do not halve the volume again.
Answer: 108\text{ cm}^3

Worked example 4: A trapezium cross-section

A prism is 12 cm long. Its cross-section is a trapezium. The parallel sides are 3 cm and 7 cm, and the perpendicular height is 4 cm.

Work out the volume of the prism.

The diagram shows the cross-section. The prism is 12 cm long.
The area of a trapezium is \frac{1}{2}(a + b)h, where a and b are the parallel sides.
\frac{1}{2} \times (3 + 7) \times 4 = \frac{1}{2} \times 10 \times 4 = 20\text{ cm}^2
Your turn. Multiply the area by the length. Work out 20 \times 12

Your answer

20 \times 12 = 240
Answer: 240\text{ cm}^3

Working backwards from a volume

Some questions give you the volume and ask for a missing length. Use the same formula, and fill in what you know.

For a prism, volume \div length gives the area of the cross-section. For example, a prism with volume 90\text{ cm}^3 and length 6 cm has a cross-section of area 90 \div 6 = 15\text{ cm}^2

That area is not the answer yet. Write down the area formula for the cross-section, put in the area and the length you know, and work out the missing one.

Worked example 5: Finding a length from a volume

A tent is a triangular prism. The tent is 5 m long and its volume is 30\text{ m}^3. The cross-section is a triangle with perpendicular height 3 m.

Work out the width, w, of the base of the triangle.

The shaded triangle is the cross-section. Its height is 3 m, and its width, w, is the length we want.
Your turn. Volume \div length gives the area of the triangle. Work out 30 \div 5

Your answer

30 \div 5 = 6, so the triangle has area 6\text{ m}^2
The area of the triangle is \frac{1}{2} \times w \times 3, so \frac{1}{2} \times w \times 3 = 6
Double both sides to remove the half: w \times 3 = 12
w = 12 \div 3 = 4
Answer: w = 4 m

Containers, litres and packing

Capacity is how much a container can hold. For liquids, use these facts:

1 \text{ litre} = 1000\text{ cm}^3 \qquad 1\text{ ml} = 1\text{ cm}^3

A tank 40 cm long, 25 cm wide and 30 cm high holds 40 \times 25 \times 30 = 30\,000\text{ cm}^3. Divide by 1000 to change to litres: 30\,000 \div 1000 = 30 litres.

If a container is only part full, use the depth of the liquid, or take the fraction of the full volume. When you pour into cups, the answer is the number of cups filled completely, so round down.

To pack small boxes into a big box, work out how many fit along each edge, then multiply the three numbers, just as you counted cubes in layers. Change all the lengths to the same unit first.

Worked example 6: A rise in water level

A fish tank is a cuboid. Its base is 30 cm long and 20 cm wide. The water in the tank is 8 cm deep. Jo pours in another 3 litres of water. No water spills out.

Work out the new depth of the water.

The new water forms a layer on top of the old water, over the same base.
Change the litres to \text{cm}^3: 3 \times 1000 = 3000\text{ cm}^3
The area of the base is 30 \times 20 = 600\text{ cm}^2
Your turn. The rise in depth is volume \div base area. Work out 3000 \div 600

Your answer

3000 \div 600 = 5, so the water rises by 5 cm.
The question asks for the new depth, so add the rise to the old depth: 8 + 5 = 13
Answer: 13 cm

Common mistakes

Working out the surface area or adding the edges. For volume, multiply length, width and height.
Halving in the wrong place. Halve only for a triangle's area; a cuboid's volume is never halved.
Stopping at the area of the cross-section. Volume \div length gives the area; use the area formula to find the missing length.
Using 10 or 100 to change to litres. 1 litre = 1000\text{ cm}^3, so divide by 1000
Rounding up the number of cups. Only cups filled completely count, so round down.
Leaving out the units. Write volumes in cubic units, such as \text{cm}^3