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GCSE Maths · Geometry and measures

Similar shapes GCSE Questions and Worked Answers

Similar shapes have equal corresponding angles and all matching lengths in one ratio. Lengths scale by k; areas by k² and volumes by k³. Area and volume similarity are Higher-tier extensions.

Foundation & Higher6 worked examples10 original questions
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Start with the meaning

What you need to know about similar shapes

When you enlarge a photograph without stretching it, every length changes by the same multiplier. A 3 cm by 2 cm picture becomes 6 cm by 4 cm when both dimensions double. The size changes, but its proportions and corner angles stay the same. The two rectangles are similar.

See the idea first

Match sides before choosing a multiplier

Corresponding sides are sides in matching positions. Divide a target length by its matching original length to find the scale factor k. Here k = 6 ÷ 3 = 2. The matching height is multiplied by that same 2, not increased by the same number of centimetres.

3 cm2 cm6 cm4 cm
Both lengths double. The large rectangle contains four copies of the small rectangle, so its area is four times as large.
From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Find a missing length

What the problem asks: Similar triangles have corresponding bases 5 cm and 15 cm. A second side of the small triangle is 4 cm. Find the matching large side.

How to solve it: The small-to-large multiplier is 15 ÷ 5 = 3. The missing side is 4 × 3 = 12 cm.

Decide whether shapes are similar

What the problem asks: Are rectangles 4 cm by 6 cm and 6 cm by 9 cm similar?

How to solve it: Both dimensions scale by 1.5 and all their angles are right angles, so yes. For general polygons, matching side ratios alone do not establish equal corresponding angles.

Scale an area or volume

What the problem asks: All lengths of a solid are tripled. What happens to area and volume?

How to solve it: Area involves two dimensions, so its multiplier is 3 × 3 = 9. Volume involves three, so its multiplier is 27. These area/volume relationships are Higher-tier work.

A reliable routine

Find an unknown corresponding length

Use one common multiplier only when similarity is given or established. Equal corresponding angles preserve shape, while a common length multiplier preserves proportions.

  1. Identify corresponding sides by matching angles and positions, not just left and right on the page.
  2. Choose a direction: original to target.
  3. Divide target known length by original matching length.
  4. Multiply another original length by the same scale factor, or divide to reverse the direction.

Check: For Higher-tier reverse questions, take a square root of an area ratio or a cube root of a volume ratio to recover the length scale factor.

Fully worked

Similar shapes GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

A matching side

2 marks
Question

Two similar triangles have corresponding sides 7 cm and 21 cm. Another side of the smaller triangle is 5 cm. Find its corresponding larger side.

k=21/7=3k=21/7=3 Larger side=5(3)=15 cm\text{Larger side}=5(3)=15\text{ cm}

Tip: use sides that are stated to correspond.

Example 2

Scale down

2 marks
Question

A model and real object are similar. A 12 cm model length corresponds to 60 cm on the object. A different real length is 35 cm. Find the model length.

Model-to-real scale is 60/12=560/12=5. Reverse by dividing:

35/5=7 cm35/5=7\text{ cm}
Example 3

Check two ratios

3 marks
Question

Are rectangles measuring 4 cm by 10 cm and 6 cm by 15 cm similar? Explain.

The matching length ratios are

6/4=1.56/4=1.5 15/10=1.515/10=1.5

All corresponding angles are 90°. The same scale factor works for both dimensions, so the rectangles are similar.

Example 4

Area scale factor

Higher only3 marks
Question

Two similar shapes have length scale factor 4 from small to large. The small area is 7 cm². Find the large area.

Two dimensions are multiplied by 4, so the area factor is

42=164^2=16 Alarge=7(16)=112 cm2A_{\rm large}=7(16)=112\text{ cm}^2

Do not multiply area by just 4.

Example 5

Recover a length from area

Higher only4 marks
Question

Similar shapes have areas 18 cm² and 162 cm². A small side is 5 cm. Find the matching large side.

Area factor=162/18=9\text{Area factor}=162/18=9 k=9=3k=\sqrt9=3 Large side=5(3)=15 cm\text{Large side}=5(3)=15\text{ cm}

Take the square root before scaling a length.

Example 6

Volume to surface area

Higher only4 marks
Question

Similar solids have volumes 16 cm³ and 128 cm³. The small solid's surface area is 30 cm². Find the large surface area.

Volume factor=128/16=8\text{Volume factor}=128/16=8 k=83=2k=\sqrt[3]8=2 Area factor=22=4\text{Area factor}=2^2=4 Slarge=30(4)=120 cm2S_{\rm large}=30(4)=120\text{ cm}^2

Volume and area use different powers of the same length factor.

10 original questions · total 22 marks

Similar shapes GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 27 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
1

Double a length

2 marks

Similar rectangles have widths 4 cm and 8 cm. The small height is 3 cm. Find the large height.

Show worked answer
k=8/4=2k=8/4=2 h=3(2)=6 cmh=3(2)=6\text{ cm}
2

Fractional enlargement

2 marks

A length of 10 cm becomes 15 cm in a similar shape. What does a matching 8 cm length become?

Show worked answer
k=15/10=1.5k=15/10=1.5 8(1.5)=12 cm8(1.5)=12\text{ cm}
3

Reverse direction

2 marks

Similar shapes have length ratio small : large = 2 : 7. A large side is 28 cm. Find the matching small side.

Show worked answer
28×27=8 cm28\times\frac27=8\text{ cm}

Use the small-to-large ratio in the correct direction.

4

Not similar

2 marks

Are rectangles 3 cm by 5 cm and 6 cm by 9 cm similar?

Show worked answer

No. One dimension doubles, but the other has multiplier 9 ÷ 5 = 1.8. Equal angles alone do not make the rectangles similar.

5

Angle correspondence

2 marks

Similar triangles ABC and DEF have A corresponding to D and B to E. If angle A is 42° and B is 68°, find angle F.

Show worked answer
C=180−42−68=70∘C=180-42-68=70^\circ

F corresponds to C, so angle F is 70°.

6

Scale perimeter

2 marks

Two similar shapes have length scale factor 3.5. The smaller perimeter is 12 cm. Find the larger perimeter.

Show worked answer

Every edge scales by 3.5, so their sum does too.

P=12(3.5)=42 cmP=12(3.5)=42\text{ cm}

Perimeter is a length measure.

7

Area ratio

Higher only2 marks

Similar shapes have length ratio 3 : 5. Find their area ratio.

Show worked answer

Square both parts:

32:52=9:253^2:5^2=9:25
8

Volume ratio

Higher only2 marks

Similar solids have length ratio 2 : 3. Find their volume ratio.

Show worked answer

Cube both parts:

23:33=8:272^3:3^3=8:27
9

Reverse volume

Higher only3 marks

Similar solids have volumes 27 cm³ and 216 cm³. A small height is 4 cm. Find the larger height.

Show worked answer
216/27=8216/27=8 k=83=2k=\sqrt[3]8=2 h=4(2)=8 cmh=4(2)=8\text{ cm}
10

Area to volume ratio

Higher only3 marks

Similar solids have surface-area ratio 16 : 25. Find their volume ratio.

Show worked answer

First recover the length ratio: 16:25=4:5\sqrt{16}:\sqrt{25}=4:5. Cube that ratio:

43:53=64:1254^3:5^3=64:125
Examiner-style feedback

Common similar shapes mistakes

Adding a fixed amount

Similarity uses a multiplier for lengths, not the same additive increase.

Using one power for everything

Length and perimeter use k; area uses k²; volume uses k³.

Comparing unmatched sides

Match corresponding angles and positions before forming ratios, especially for rotated triangles.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

  1. Establish similarity.
  2. Match corresponding lengths.
  3. Keep a clear scaling direction.
  4. Use the power appropriate to the measurement.
Quick answers

Similar shapes FAQ

Are all rectangles similar?

No. Their length-to-width ratios must match, allowing for rotation.

Is congruence the same as similarity?

Congruent shapes have the same shape and size: similarity with length scale factor 1. Similar shapes may have different sizes.

Build connected skills

What to revise next

Browse all GCSE Maths study guidesReview the worked examplesTry the questions again
Content standards

Curriculum and rights review

G19/R12: corresponding lengths across tiers, with area/volume scale factors and reverse-root applications labelled Higher. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references