Similar shapes: area and volume

When a shape is enlarged by scale factor k, lengths multiply by k but areas multiply by k2 and volumes by k3.

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The three scale factors

Length scale factor k.
Area scale factor k2 (area has two dimensions).
Volume scale factor k3 (volume has three dimensions).
Lengths 2 : 5 means areas 4 : 25 and volumes 8 : 125.
LengthAreaVolume
Scale factorkk2k3
Example k = 3ร— 3ร— 9ร— 27

Working backwards

Areas 16 : 36. Square root: lengths 4 : 6 = 2 : 3.
Volumes 8 : 27. Cube root: lengths 2 : 3, so areas 4 : 9.
Always go back to the length scale factor first, then go where you need.

Frustums

A frustum is a cone with the top cut off. Volume = big cone โˆ’ small cone.
A cone of height 12 cm and radius 6 cm has the top 6 cm cut off. The small cone is similar with k = 12, so its radius is 3 cm.
Frustum = 13ฯ€(62)(12) โˆ’ 13ฯ€(32)(6) = 126ฯ€ = 395.8 cm3
Worked example

Two similar solids have surface areas 16 cm2 and 36 cm2. The smaller has volume 40 cm3. Find the volume of the larger.

  1. Area factor 3616 = 94, so length factor 32.
  2. Volume factor (32)3 = 278
  3. 40 ร— 278 = 135 cm3

Answer: 135 cm3

Key idea

Length ร— k, area ร— k2, volume ร— k3. To go from area to volume (or back), find k first with a square root or a cube root.

Key words
Similar
The same shape but a different size: angles are equal and lengths are in the same ratio.
Scale factor
The number every length is multiplied by in an enlargement or between similar shapes.
Radius
The distance from the centre of a circle to its edge.

All GCSE and International GCSE Mathematics key terms and definitions โ†’

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  1. Two similar shapes have lengths in the ratio 1 : 4. The smaller has area 2 cm2. Find the area of the larger.

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    Answer: 32 cm2

    Area factor 42 = 16. 2 ร— 16 = 32.

  2. Two similar vases have heights 6 cm and 9 cm. The smaller holds 240 ml. How much does the larger hold?

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    Answer: 810 ml

    k = 96 = 32. Volume factor 278. 240 ร— 278 = 810.

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Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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