Find the scale factor between two similar shapes from a matching pair of sides, then multiply or divide to find missing lengths. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page.
Similar shapes are the same shape, possibly different size. Matching angles are equal; matching sides are in the same ratio (the scale factor). Scale factor =old lengthnew length. Multiply by it to enlarge; divide to reduce. Corresponding sides are opposite corresponding angles — sketch and mark them.
A missing length in two similar triangles is often a pair of overlapping triangles. Write the matching sides as a fraction equal to another matching pair, then cross-multiply. Length scale factor only — area and volume scale factors are the next grade.
Matching sides are in the same ratio (scale factor). Multiply a small length by the scale factor to get the large; divide to go large to small. Matching angles are equal — that is how you know which sides match.
Objectives
Enlarge a length by a whole-number scale factor.
Shrink a length by a fractional scale factor.
Find the scale factor from two sides, and use a map scale.
Key points
Identify corresponding sides in two similar shapes, even when one is rotated or flipped.
Find the scale factor by dividing a length on the large shape by the matching length on the small shape.
Multiply by the scale factor to go small to large, divide to go large to small.
Handle nested triangles with parallel lines, where the whole side (not just the extra part) is used.
Worked examples
Example 1
A shape is enlarged by length scale factor 3. A side of 4 cm becomes what length?
Slip: Conrad added 100 to 3 Ratios represent multiplication.
Slip: Chloe divided 100 by 3 Real length is 100 times the map distance.
Slip: Conrad wrote 103. The scale number multiplies the map length: 3 × 100 = 300.
Questions and answers
GCSE Maths similar shapes: lengths questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
A shape is enlarged by length scale factor 4. A side of 5 cm becomes what length?
Show answer
Answer: 20
Multiply side by scale factor: 4 × 5.
Result is 20.
M1 — Multiply side by scale factor: 4 × 5.
A1 — Result is 20.
Question 2 Easy2 marks
A shape is enlarged by length scale factor 2. A side of 7 cm becomes what length?
Show answer
Answer: 14
Multiply side by scale factor: 2 × 7.
Result is 14.
M1 — Multiply side by scale factor: 2 × 7.
A1 — Result is 14.
Question 3 Medium2 marks
A shape is shrunk by length scale factor 1/2. A side of 8 cm becomes what length?
Show answer
Answer: 4
Multiply side by scale factor: (1/2) × 8.
Result is 4.
M1 — Multiply side by scale factor: (1/2) × 8.
A1 — Result is 4.
Question 4 Medium2 marks
A shape is shrunk by length scale factor 1/3. A side of 9 cm becomes what length?
Show answer
Answer: 3
Multiply side by scale factor: (1/3) × 9.
Result is 3.
M1 — Multiply side by scale factor: (1/3) × 9.
A1 — Result is 3.
Question 5 Hard2 marks
Two similar shapes have matching sides of 4 cm and 12 cm. Work out the scale factor from the smaller shape to the larger shape.
Show answer
Answer: 3
Divide larger by smaller: 12 / 4.
Result is 3.
M1 — Divide larger by smaller: 12 / 4.
A1 — Result is 3.
Question 6 Hard2 marks
Two similar shapes have matching sides of 6 cm and 18 cm. Work out the scale factor from the smaller shape to the larger shape.
Show answer
Answer: 3
Divide larger by smaller: 18 / 6.
Result is 3.
M1 — Divide larger by smaller: 18 / 6.
A1 — Result is 3.
Question 7 Exam-style2 marks
Two similar shapes have matching sides of 2 cm and 8 cm. Work out the scale factor from the smaller shape to the larger shape.
Show answer
Answer: 4
The smaller side is 2 and the larger side is 8.
Divide larger by smaller: 8 / 2 = 4.
M1 — 8 ÷ 2
A1 — 4
Common mistakes
Most lost marks on this topic are one slip.
Adding the scale factor instead of multiplying
Scale factors are multiplicative: always multiply lengths by k
Dividing when enlarging or multiplying when finding map ratio backwards
From map to real life, multiply by scale number
Inverting the ratio when finding scale factor from smaller
Scale factor from smaller to larger is larger ÷ smaller (> 1)
Spot the slip
A shape is enlarged by length scale factor 3. A side of 4 cm is on the shape. Ada adds 3 instead of multiplying. What is the correct new length?
12
7
4
Show the slip
12 Multiply the side by the scale factor.
Exam tips
This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
State the scale factor as its own line — it is almost always a method mark.
If a question asks 'explain why the triangles are similar', mention the equal angles (parallel lines give corresponding angles, and the shared angle at the apex).
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.