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GCSE Maths / Edexcel

Similar shapes and congruence

Identify congruent and similar shapes, use linear scale factors for corresponding lengths, and handle area scale factors for Higher questions.

Geometry and MeasuresFoundation and HigherGrades 4 to 8Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Similar shapes and congruence

Pearson Edexcel GCSE Maths geometry G5 and G6: use congruence and similarity, including scale factors and corresponding lengths.

Practise this topicUpload your answerAsk AI to explain thisTry exam-style questions
Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

Congruent shapes are exactly the same shape and size. They may be rotated, reflected or translated, but all matching sides and angles are equal.

Similar shapes have the same shape but not necessarily the same size. Matching angles are equal and matching sides are in the same ratio.

The linear scale factor tells you how to multiply a length on one shape to get the matching length on the other shape.

To find the scale factor, divide a known length on the image by the matching known length on the original shape.

Corresponding sides must match the same positions. Do not compare a base with a height unless they correspond.

If the scale factor is greater than 1, the shape is enlarged. If it is between 0 and 1, the shape is reduced.

Higher: area scale factor is the square of the linear scale factor. If lengths double, areas multiply by 4.

Congruence is about same size. Similarity is about same shape with a possible scale factor.

Key ruleSimilar shapes have equal matching angles and proportional matching sides. Congruent shapes have equal matching sides and angles.

Diagram guide

4 cm4 cm8 cm8 cmsmallsimilar shapescale factor = 8 ÷ 4 = 2same angles, corresponding sides in the same ratio
Similar shapesMatching side lengths are connected by the same scale factor.
congruentsame size and shapesimilarsame shape, different sizecongruent: matching sides equalsimilar: matching sides have a scale factor
Congruent or similarCongruent means same shape and size. Similar means same shape but possibly different size.

Worked examples

Find a scale factor

Two similar triangles have matching sides 4 cm and 10 cm. Find the scale factor from the smaller triangle to the larger triangle.

4 cm4 cm8 cm8 cmsmallsimilar shapescale factor = 8 ÷ 4 = 2same angles, corresponding sides in the same ratio
Example: linear scale factorDivide the larger matching length by the smaller matching length.
  1. Scale factor = larger length / smaller length.
  2. 10 / 4 = 2.5.

Answer: Scale factor 2.5

Find a missing side

Two similar shapes have scale factor 3 from small to large. A side on the small shape is 7 cm. Find the matching side on the large shape.

4 cm4 cm8 cm8 cmsmallsimilar shapescale factor = 8 ÷ 4 = 2same angles, corresponding sides in the same ratio
Example: multiply by scale factorSmall to large means multiply by 3.
  1. Matching side on large shape = 7 x 3.
  2. 7 x 3 = 21.

Answer: 21 cm

Congruent or similar

Two triangles have matching sides 5 cm, 7 cm, 9 cm and 5 cm, 7 cm, 9 cm. Are they congruent or just similar?

congruentsame size and shapesimilarsame shape, different sizecongruent: matching sides equalsimilar: matching sides have a scale factor
Example: congruent trianglesSame matching side lengths means same size and shape.
  1. All matching side lengths are equal.
  2. The triangles are the same size and shape.

Answer: Congruent

Higher: area scale factor

Two similar shapes have linear scale factor 3. The area of the smaller shape is 8 cm². Find the area of the larger shape.

4 cm4 cm8 cm8 cmsmallsimilar shapescale factor = 8 ÷ 4 = 2same angles, corresponding sides in the same ratio
Example: area scale factorArea scale factor is the square of the length scale factor.
  1. Area scale factor = 3² = 9.
  2. Larger area = 8 x 9.
  3. 8 x 9 = 72.

Answer: 72 cm²

Common mistakes

  • Comparing non-matching sides.
  • Adding the scale factor instead of multiplying.
  • Thinking similar means exactly the same size.
  • Calling shapes congruent when one is an enlargement of the other.
  • Higher: using the linear scale factor for area instead of squaring it.

Quick exercise

Try these before moving to the exam-style questions.

  1. Two matching sides are 6 cm and 18 cm. Find the scale factor from small to large.
    4 cm4 cm8 cm8 cmsmallsimilar shapescale factor = 8 ÷ 4 = 2same angles, corresponding sides in the same ratio
    Quick check: scale factorDivide large by small.
  2. A side is 5 cm on a small shape. Scale factor to the large shape is 4. Find the matching side.
    4 cm4 cm8 cm8 cmsmallsimilar shapescale factor = 8 ÷ 4 = 2same angles, corresponding sides in the same ratio
    Quick check: missing lengthMultiply by the scale factor.
  3. What word means same shape and same size?
    congruentsame size and shapesimilarsame shape, different sizecongruent: matching sides equalsimilar: matching sides have a scale factor
    Quick check: same sizeCongruent shapes can be moved or turned, but not resized.
  4. What word means same shape but possibly different size?
    congruentsame size and shapesimilarsame shape, different sizecongruent: matching sides equalsimilar: matching sides have a scale factor
    Quick check: same shapeSimilar shapes have proportional matching sides.
  5. Higher: linear scale factor is 2. What is the area scale factor?
    4 cm4 cm8 cm8 cmsmallsimilar shapescale factor = 8 ÷ 4 = 2same angles, corresponding sides in the same ratio
    Quick check: area scale factorSquare the linear scale factor.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleFoundationNon-calculator2 marks

Two similar rectangles have matching sides 3 cm and 12 cm. Find the scale factor from the smaller rectangle to the larger rectangle.

4 cm4 cm8 cm8 cmsmallsimilar shapescale factor = 8 ÷ 4 = 2same angles, corresponding sides in the same ratio
Question diagram: matching sidesUse corresponding sides to find the scale factor.
similar shapesscale factorfoundation geometry
Standard exam styleFoundation and HigherCalculator3 marks

Two similar triangles have scale factor 2.5 from A to B. A side on triangle A is 8 cm. Find the matching side on triangle B.

4 cm4 cm8 cm8 cmsmallsimilar shapescale factor = 8 ÷ 4 = 2same angles, corresponding sides in the same ratio
Question diagram: missing sideA to B uses multiplication by 2.5.
similar trianglesmissing sidescale factor
ChallengeHigherCalculator4 marks

Two similar shapes have linear scale factor 1.5 from small to large. The area of the smaller shape is 40 cm². Find the area of the larger shape.

4 cm4 cm8 cm8 cmsmallsimilar shapescale factor = 8 ÷ 4 = 2same angles, corresponding sides in the same ratio
Question diagram: area scale factorFor areas, square the linear scale factor.
similar areasarea scale factorhigher geometry