Similarity: lengths, areas and volumes · Higher notes

Similar shapes have corresponding lengths in a common ratio. Areas involve two length dimensions, so their scale factor is the square of the length factor. Volumes involve three dimensions, so their scale factor is the cube. This explains why doubling every length does not merely double the area or volume. When you start with an area or volume ratio, work backwards using a square root or cube root before scaling a different type of measurement. Match the larger and smaller shapes consistently.

Learning objectives

Convert a length scale factor into area and volume factors.
Work backwards from an area or volume ratio to a length ratio.

Key ideas

For similar shapes, use corresponding measurements and a consistent comparison direction.

For a positive length scale factor, square it for area and cube it for volume.

k>0,\qquad\text{area factor}=k^2,\quad\text{volume factor}=k^3

Take square roots of an area ratio to recover the length ratio.

a:b\quad\Rightarrow\quad\sqrt a:\sqrt b

Take cube roots of a volume ratio to recover the length ratio.

a:b\quad\Rightarrow\quad\sqrt[3]a:\sqrt[3]b

Worked examples

Example 1

In two separate similarity examples, the small-to-large length ratio is 2 to 3. One small shape has area 24 square centimetres; a different small solid has volume 80 cubic centimetres. Find the corresponding large area and volume.

k=\frac32

A_{\text{large}}=24\left(\frac32\right)^2

V_{\text{large}}=80\left(\frac32\right)^3

Answer

A_{\text{large}}=54\ \text{cm}^2,\qquad V_{\text{large}}=270\ \text{cm}^3

Example 2

Similar solids have volume ratio 27 to 64. Find the length ratio.

\sqrt[3]{27}:\sqrt[3]{64}

Answer

3:4

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Calculate corresponding length, area or volume scale factors.

AO2 · Reasoning

Explain why area and volume scale factors differ from the length scale factor.

AO3 · Problem solving

Work backwards from an area or volume ratio to a missing dimension.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

Do not apply the length factor directly to areas or volumes. Keep small-to-large and large-to-small comparisons consistent.

Try it yourself

Attempt each question before opening its answer.

Check 1: Corresponding lengths increase by factor 3. What is the area factor?

Answer: 9

The area factor is 3².

Check 2: Similar solids have volume ratio 27:64. Find the length ratio.

Answer: 3:4

Take cube roots of both parts.

Explain it in your own words

Explain geometrically why area scales by k².
Write how to recover a length factor from a volume factor.
Show how an area ratio 49:81 leads to a volume ratio.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes

Specification source

AQA Geometry specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.