Similar Shapes and Scale Factors –

Similar shapes have the same shape but different sizes — all corresponding angles are equal and corresponding sides are in the same ratio. This topic spans Foundation and Higher GCSE Maths, with Higher students needing to handle area and volume scale factors. It appears regularly on AQA, Edexcel, and OCR exams. This guide explains linear, area, and volume scale factors with worked examples and practice questions.

What Are Similar Shapes?

Two shapes are similar if one is an enlargement of the other. This means all corresponding angles are equal and all corresponding lengths are in the same ratio. That ratio is called the linear scale factor (k).

Key Formulas

If the linear scale factor from shape A to shape B is k, then areas are multiplied by k² and volumes by k³.

Step-by-Step Method

1. Identify corresponding sides by matching angles or using the diagram.
2. Calculate the linear scale factor by dividing a known side of one shape by the corresponding side of the other.
3. Find missing lengths by multiplying or dividing by k.
4. For area problems (Higher), square the linear scale factor to get the area scale factor.
5. For volume problems (Higher), cube the linear scale factor to get the volume scale factor.

Worked Example 1 — Foundation Level

Question: Triangles A and B are similar. Triangle A has sides 4 cm, 6 cm, and 8 cm. The shortest side of triangle B is 10 cm. Find the other two sides of triangle B.

Working:

The shortest side of A is 4 cm and the shortest side of B is 10 cm.

Linear scale factor k = 10 ÷ 4 = 2.5

Second side of B = 6 × 2.5 = 15 cm

Third side of B = 8 × 2.5 = 20 cm

Answer: The sides of triangle B are 10 cm, 15 cm, and 20 cm.

Worked Example 2 — Higher Level

Question: Two similar cylinders have heights 6 cm and 15 cm. The surface area of the smaller cylinder is 120 cm². Find the surface area of the larger cylinder.

Working:

Linear scale factor k = 15 ÷ 6 = 2.5

Area scale factor = k² = 2.5² = 6.25

Surface area of larger cylinder = 120 × 6.25 = 750 cm²

Answer: The surface area of the larger cylinder is 750 cm².

Worked Example 3 — Exam Style

Question: Two similar solid cones have volumes 54 cm³ and 432 cm³. The height of the smaller cone is 6 cm. Find the height of the larger cone.

Working:

Volume scale factor = 432 ÷ 54 = 8

Linear scale factor k = cube root of 8 = 2

Height of larger cone = 6 × 2 = 12 cm

Answer: The height of the larger cone is 12 cm.

Common Mistakes

• Using the linear scale factor for area or volume. If k = 3, the area scale factor is 9 (not 3) and the volume scale factor is 27 (not 3). Students frequently forget to square or cube.
• Matching the wrong sides. Always pair the shortest side with the shortest side, or use the diagram to identify corresponding sides by their position relative to equal angles.
• Dividing instead of multiplying (or vice versa). If going from small to large, multiply by k. If going from large to small, divide by k.

Exam Tips

• Write down the scale factor clearly before doing any further calculations — this avoids errors in multi-step problems.
• For area and volume problems, it can help to write: "Linear SF = ..., Area SF = (...)², Volume SF = (...)³" as a checklist.
• If the question gives you areas, find the area scale factor first, then square root it to get the linear scale factor.
• If the question gives you volumes, find the volume scale factor first, then cube root it to get the linear scale factor.
• Real-world contexts (maps, models, packaging) often appear — the same rules apply.

Practice Questions

Q1 (Foundation): Two similar rectangles have widths 3 cm and 9 cm. The length of the smaller rectangle is 5 cm. Find the length of the larger rectangle.

Q2 (Foundation): Two similar triangles have corresponding sides of 8 cm and 12 cm. Find the linear scale factor from the smaller to the larger.

Q3 (Higher): Two similar prisms have surface areas of 50 cm² and 200 cm². The volume of the smaller prism is 80 cm³. Find the volume of the larger prism.

Practise similar shapes and scale factors with instant feedback free on GCSEMathsAI.

Related Topics

• Congruent Triangles
• Congruence and Similarity
• Enlargement and Centre of Enlargement

Summary

• Similar shapes have equal corresponding angles and sides in the same ratio. The linear scale factor k gives the ratio of corresponding lengths. The area scale factor is k² and the volume scale factor is k³. To find missing lengths, multiply or divide by k. For area or volume, use k² or k³ respectively. Always identify corresponding sides carefully and state your scale factor before calculating.