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Geometry and measuresFoundation and Higher (some Higher only)

Similar shapes GCSE questions with answers

8 similar shapes questions to try, each with the answer and the full working one tap away. Every question here was made and worked out by the Show Working maths engine, not typed in by hand.

Similar shapes have the same angles, and every length in one is the same number of times bigger than in the other. You use this scale factor to find missing lengths, and on Higher to find areas and volumes of similar shapes.

Quiz yourself: 8 questions

Worked example 1: similar shapes

Triangles A and B are similar. Work out the length marked x.

10 cm15 cm7 cmxAB
Not drawn accurately
  1. Similar shapes are enlargements of each other. Find the scale factor from the pair of sides you know: 15 ÷ 10 = 1.5.
  2. x is on the bigger triangle, so multiply: 7 × 1.5 = 10.5 cm.

Answer 10.5 cm

01 / Quiz yourselfTry it, then tap to check

Similar shapes questions to try

Work each one out on paper first. Then open the answer and compare your working with ours, step by step.

01 / Similar shapesBoth tiers

Triangles A and B are similar. Work out the length marked x.

10 cm25 cmx35 cmAB
Not drawn accurately
Show the answer and working
  1. Similar shapes are enlargements of each other. Find the scale factor from the pair of sides you know: 25 ÷ 10 = 2.5.
  2. x is on the smaller triangle, so divide: 35 ÷ 2.5 = 14 cm.

Answer 14 cm

02 / Similar shapesBoth tiers

Triangles A and B are similar. Work out the length marked x.

12 m30 mx17.5 mAB
Not drawn accurately
Show the answer and working
  1. Similar shapes are enlargements of each other. Find the scale factor from the pair of sides you know: 30 ÷ 12 = 2.5.
  2. x is on the smaller triangle, so divide: 17.5 ÷ 2.5 = 7 m.

Answer 7 m

03 / Similar shapesBoth tiers

Triangles A and B are similar. Work out the length marked x.

18 cm27 cmx15 cmAB
Not drawn accurately
Show the answer and working
  1. Similar shapes are enlargements of each other. Find the scale factor from the pair of sides you know: 27 ÷ 18 = 1.5.
  2. x is on the smaller triangle, so divide: 15 ÷ 1.5 = 10 cm.

Answer 10 cm

04 / Similar shapesBoth tiers

Triangles A and B are similar. Work out the length marked x.

16 cm64 cmx24 cmAB
Not drawn accurately
Show the answer and working
  1. Similar shapes are enlargements of each other. Find the scale factor from the pair of sides you know: 64 ÷ 16 = 4.
  2. x is on the smaller triangle, so divide: 24 ÷ 4 = 6 cm.

Answer 6 cm

05 / Areas and volumes of similar shapesHigher only

Cylinders A and B are mathematically similar. A has height 4 m and B has height 10 m. The surface area of A is 48 m². Work out the surface area of B.

4 mA10 mB
Not drawn accurately
Show the answer and working
  1. The length scale factor from A to B is 10 ÷ 4 = 52.
  2. Areas scale by the square of the length scale factor: 52² = 254.
  3. Surface area of B = 48 × 254 = 300 m².

Answer 300 m²

06 / Areas and volumes of similar shapesHigher only

Cones A and B are mathematically similar. A has height 4 m and B has height 8 m. The surface area of B is 100 m². Work out the surface area of A.

4 mA8 mB
Not drawn accurately
Show the answer and working
  1. The length scale factor from A to B is 8 ÷ 4 = 2.
  2. Areas scale by the square of the length scale factor: 2² = 4.
  3. Going from B to A, divide: 100 ÷ 4 = 25 m².

Answer 25 m²

07 / Areas and volumes of similar shapesHigher only

Cylinders A and B are mathematically similar. The volume of A is 8 cm³ and the volume of B is 216 cm³. The height of A is 3 cm. Work out the height of B.

3 cmAxB
Not drawn accurately
Show the answer and working
  1. Volume scale factor from A to B: 216 ÷ 8 = 27.
  2. Volumes scale by the cube of the length scale factor, so take the cube root: ∛27 = 3.
  3. Height of B = 3 × 3 = 9 cm.

Answer 9 cm

08 / Areas and volumes of similar shapesHigher only

Cylinders A and B are mathematically similar. The surface area of A is 6 cm² and the surface area of B is 96 cm². The volume of A is 1 cm³. Work out the volume of B.

AB
Not drawn accurately
Show the answer and working
  1. Area scale factor: 96 ÷ 6 = 16.
  2. Square root it for the length scale factor: √16 = 4.
  3. Cube it for the volume scale factor: 4³ = 64.
  4. Volume of B = 1 × 64 = 64 cm³.

Answer 64 cm³

02 / What's covered2 kinds of question

Similar shapes at GCSE and where marks go

  • Similar shapesBoth tiers
  • Areas and volumes of similar shapesHigher only

Worked example 2: areas and volumes of similar shapes

Cones A and B are mathematically similar. The volume of A is 2 cm³ and the volume of B is 54 cm³. The surface area of A is 10 cm². Work out the surface area of B.

AB
Not drawn accurately
  1. Volume scale factor: 54 ÷ 2 = 27.
  2. Cube root it for the length scale factor: ∛27 = 3.
  3. Square it for the area scale factor: 3² = 9.
  4. Surface area of B = 10 × 9 = 90 cm².

Answer 90 cm²

A common mistake

Using the length scale factor for area or volume. If the lengths are 3 times bigger, the areas are 3² = 9 times bigger and the volumes 3³ = 27 times bigger.

03 / Practise itiPhone and iPad

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More geometry and measures topics

Angles and bearingsArea and perimeterCirclesVolume and surface areaPythagorasTrigonometryConverting unitsSine and cosine rulesCircle theoremsCongruent trianglesVectorsTransformationsAll GCSE maths topics
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