Similarity: lengths, areas and volumes · Revision guide

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

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}

3:4

Watch out

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