Area and Volume

What you will learn

• Say why area uses square units and volume uses cubic units.
• Find the volume of a cuboid or cube.
• Link cubic centimetres to millilitres and litres.

Before you start: you should know how to find the area of a rectangle and how to multiply three numbers. See Area and Perimeter.

The idea

Area measures a flat surface and is counted in squares: square centimetres ($\text{cm}^2$), square metres ($\text{m}^2$) and so on. A square centimetre is a square with sides of $1$ cm.

Volume measures how much three-dimensional space something takes up. It is counted in cubes: cubic centimetres ($\text{cm}^3$), cubic metres ($\text{m}^3$) and so on. A cubic centimetre is a cube with edges of $1$ cm.

Volume of a cuboid

Fill the bottom layer of a box with unit cubes: that is $l \times w$ cubes. Stack $h$ layers and you get

$V = l \times w \times h.$

A cube is a cuboid with all three lengths equal, so $V = a^3$.

Volume and capacity

Capacity is how much a container holds, usually in millilitres or litres. The link is simple: $1 \text{ cm}^3 = 1$ ml, so $1000 \text{ cm}^3 = 1$ litre. A cube $10$ cm on each side holds exactly $1$ litre.

Changing square and cubic units

Worked examples

Example 1: Cuboid volume

A box measures $20 \text{ cm} \times 15 \text{ cm} \times 10 \text{ cm}$. Find its volume and say how many litres it holds.

1. $V = 20 \times 15 \times 10 = 3000 \text{ cm}^3$.
2. $3000 \div 1000 = 3$ litres.

Answer: $3000 \text{ cm}^3$, which is $3$ litres.

Example 2: Missing height

A cuboid has a volume of $180 \text{ cm}^3$ and a base of $9 \text{ cm} \times 5 \text{ cm}$. Find its height.

1. Base area: $9 \times 5 = 45 \text{ cm}^2$.
2. Height $= 180 \div 45 = 4$ cm.

Answer: $4$ cm

Example 3: Converting square units

How many square centimetres are there in $1 \text{ m}^2$?

1. A square metre is $100 \text{ cm} \times 100 \text{ cm}$.
2. $100 \times 100 = 10\,000$.

Answer: $10\,000 \text{ cm}^2$, not $100$. For cubic units, $1 \text{ m}^3 = 1\,000\,000 \text{ cm}^3$.

Tip: before multiplying, convert all lengths to the same unit. Mixing metres and centimetres is the most common cause of wildly wrong volumes.

Why it matters

Area and volume decide how much paint, concrete, soil, water or packaging you need, and so how much things cost. Delivery companies price parcels by volume, and swimming pools are treated according to their capacity. For the wrapping of a box, see Surface Area of Cubes and Cuboids.

Practice

Your turn. Type or choose an answer, then press Check. Getting one wrong is fine, that is what practice is for. You can try as many times as you like.

Question 1 of 8

A rectangle is $8$ cm by $6$ cm. Find its area in $\text{cm}^2$.

Your answer

Check

Question 2 of 8

A cuboid is $5$ cm long, $4$ cm wide and $3$ cm high. Find its volume in $\text{cm}^3$.

Your answer

Check

Question 3 of 8

A cube has sides of $4$ cm. Find its volume in $\text{cm}^3$.

Your answer

Check

Question 4 of 8

How many square centimetres are in $1$ square metre?

Your answer

Check

Question 5 of 8

A box is $20$ cm by $15$ cm by $10$ cm. How many litres does it hold? ($1000 \text{ cm}^3 = 1$ litre.)

Your answer

Check

Question 6 of 8

A cuboid has volume $120 \text{ cm}^3$, length $6$ cm and width $5$ cm. Find its height in cm.

Your answer

Check

Question 7 of 8

A cuboid is $2.5$ m by $2$ m by $1.2$ m. Find its volume in $\text{m}^3$.

Your answer

Check

Question 8 of 8

A fish tank is $50$ cm long, $40$ cm wide and $30$ cm high. How many litres of water fill it?

Your answer

Check