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Volumes of Cubes and Cuboids

A cuboid is the box shape you meet everywhere: cereal packets, bricks, rooms. Its volume comes from one tidy rule, and once you have it, cubes and missing lengths fall out of the same idea.

MYP 5Standard MathsMensurationCriteria A · C · D~9 min read

Written by Fayek Jawad Haider · MYP student

What volume means

Volume is the amount of space a solid takes up. A neat way to picture it: how many unit cubes of side 1 cm would fit inside with no gaps? That count is the volume, measured in cubic units.

Volume
The amount of three-dimensional space a solid fills, measured in cubic units such as cm\(^3\) or m\(^3\).
Cuboid
A solid with six rectangular faces, where every corner is a right angle. A cube is the special case where all edges are equal.

The cuboid rule

Layer the unit cubes and the pattern is clear: the bottom layer holds \(l \times w\) cubes, and there are \(h\) layers stacked up. Multiply the three edge lengths together:

\[ V = l \times w \times h \]

A cube has all three edges the same length \(s\), so the rule simplifies to \(V = s \times s \times s = s^3\).

Worked example

A storage box measures 6 cm long, 4 cm wide and 5 cm high. Find its volume.

1
Write the rule: \(V = l \times w \times h\).
2
Substitute \(l = 6\), \(w = 4\) and \(h = 5\): \(V = 6 \times 4 \times 5\).
3
Multiply in steps: \(6 \times 4 = 24\), then \(24 \times 5 = 120\).
\(V = 120\) cm\(^3\)

The order does not matter

Because multiplication can be done in any order, it does not matter which edge you call length, width or height. Pick whichever pairing makes the arithmetic easiest, then multiply by the third.

Keeping units honest

Every edge must be in the same unit before you multiply. If a box is given as 20 cm by 0.5 m by 30 cm, convert the odd one out first: \(0.5\) m is \(50\) cm. Only then multiply. Because you multiplied three lengths, the answer is in cubic units. A common slip is writing cm\(^2\) instead of cm\(^3\); the little \(3\) tells the reader you found a volume, not an area.

Finding a missing length

Sometimes you are told the volume and asked for a missing edge. Rearrange the rule by dividing. If \(V = l \times w \times h\), then any one edge equals the volume divided by the product of the other two.

Worked example

A cuboid has a volume of 90 cm\(^3\). Its base is 5 cm by 6 cm. Find its height.

1
Base area: \(5 \times 6 = 30\) cm\(^2\).
2
Rearrange \(V = \text{base} \times h\) to give \(h = V \div \text{base}\).
3
Divide: \(h = 90 \div 30 = 3\).
Height \(= 3\) cm

Check yourself

1. Find the volume of a cube with edge 7 cm. +

\(V = s^3 = 7^3 = 7 \times 7 \times 7 = 343\). So \(V = \mathbf{343}\) cm\(^3\).

2. A cuboid is 8 cm by 3 cm by 2.5 cm. Find its volume. +

\(V = l \times w \times h = 8 \times 3 \times 2.5\). First \(8 \times 3 = 24\), then \(24 \times 2.5 = 60\). So \(V = \mathbf{60}\) cm\(^3\).

3. A cuboid has volume 120 cm\(^3\) and a 4 cm by 5 cm base. Find its height. +

Base area \(= 4 \times 5 = 20\) cm\(^2\). Height \(= V \div \text{base} = 120 \div 20 = \mathbf{6}\) cm.

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