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# Similar shapes
- URL: https://www.esheets.io/similar-shapes/
- Published: 2026-10-03T15:47:09.000Z
- Updated: 2026-10-03T15:47:09.000Z
- Author: Richard Linnington
- Tags: Maths, Geometry and Shape

Practise area, surface area and volume scale factors for similar shapes at GCSE Higher level. Corresponding lengths scale by **k**, areas and surface areas by **k²**, and volumes by **k³**.

Work forwards from lengths or backwards from areas and volumes: square-root an area factor or cube-root a volume factor to recover k. Keep the direction consistent, from A to B or from B to A, throughout each calculation. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Eight questions, worth one mark each, build from area and volume scale factors to mixed similar-solid problems. A question with several entries earns its mark when all entries are correct.

Start with k² and k³, then find a missing area, recover a length from areas, find a volume and recover a length from volumes. Finish by finding a volume from surface areas, converting ratios, and finding a surface area from a volume ratio.

#### What you’ll practise

- Direct and reverse area and volume scaling, including enlargements and reductions.
- Square roots and cube roots to recover the length scale factor.
- Surface-area-to-volume problems and length, surface-area and volume ratio conversion.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked examples.

## Similar shapes

Answer each question and press Check answer. Give exact values and ratios in simplest integer form; diagrams are not drawn to scale.

## Topic guide

### Choose the right scale factor

Similar shapes have corresponding lengths in a constant ratio. If the length scale factor from A to B is k, every length is multiplied by k, every area or surface area by k², and every volume by k³. You do not need an area or volume formula when that quantity is already given.

Use corresponding lengths to find k = length on B ÷ length on A. For a reduction, k is less than 1\. Keep A-to-B or B-to-A consistent throughout the question; diagrams need not be drawn exactly to scale.

### Worked area example

Two similar shapes have corresponding lengths 8 cm and 12 cm. The smaller shape has area 28 cm². From smaller to larger, k = 12 ÷ 8 = 3/2\. The area factor is (3/2)² = 9/4, so the larger area is 28 × 9/4 = **63 cm²**.

### Work backwards from an area or volume

If areas or surface areas are given, first find their factor, then take its **square root** to recover k. If volumes are given, take the **cube root** of the volume factor instead.

For example, similar solids A and B have volumes 135 cm³ and 40 cm³. A corresponding length on A is 9 cm. From A to B, the volume factor is 40/135 = 8/27\. So k = ∛(8/27) = 2/3 and the length on B is 9 × 2/3 = **6 cm**.

### Move between surface area and volume

If the surface-area ratio A:B is 16:25, square-root both terms to get the length ratio 4:5\. Cube those terms to get the volume ratio **64:125**. If A has volume 192 cm³, B has volume 192 × 125/64 = **375 cm³**. In the other direction, cube-root the volume ratio and then square the length ratio to find the surface-area ratio.

### Common mistakes

- Multiplying area by k instead of k².
- Multiplying volume by k² instead of k³.
- Square-rooting a volume ratio or cube-rooting an area ratio.
- Reversing the scale factor part-way through a calculation.
- Confusing length in cm, area in cm² and volume in cm³.
- Measuring a diagram or assuming it is drawn exactly to scale.

Before calculating, identify what is known, what is wanted and the direction of scaling.