GCSE Maths Similar Shapes & Congruence Notes
Similar shapes have the same angles and proportional sides. Length, area and volume each scale by a different power of the scale factor.
Key concepts
- Similar
- Same shape, different size — corresponding angles equal and corresponding sides in the same ratio.
- Congruent
- Identical in shape and size. Proven by SSS, SAS, ASA or RHS.
- Scale factor (SF)
- The ratio of corresponding lengths. Find it by dividing a new length by the matching old length.
Key formulas
Area scale factor
area SF = (length SF)²
Volume scale factor
volume SF = (length SF)³
Worked example
Two similar solids have length SF 3. If the small one has volume 20 cm³, find the large volume.
- 1Volume scale factor = (length SF)³ = 3³ = 27.
- 2Multiply the small volume by 27: 20 × 27.
- 3Calculate: 540 cm³.
Answer: 540 cm³
Common mistakes
✗ Using the length scale factor for area or volume.
✓ Square it for area, cube it for volume.
✗ Matching non-corresponding sides.
✓ Line up the shapes by equal angles so you compare matching sides.
✗ Assuming equal angles means congruent.
✓ Equal angles only prove similar; congruence needs matching side lengths too.
Quick check
Try these, then reveal the answers.
Length SF is 4. What is the area SF?Reveal answer
16
Which rule proves a right-angled triangle congruent using hypotenuse?Reveal answer
RHS
Similar rectangles: 4 cm side maps to 10 cm. SF?Reveal answer
2.5
Exam tips
- Always find the length scale factor first, then square or cube it as needed.
- For congruence proofs, quote the exact condition (SSS, SAS, ASA or RHS).
- Decide whether you are scaling up (SF > 1) or down (SF < 1) before dividing or multiplying.
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Frequently asked questions
What is the difference between similar and congruent?+
Similar shapes have the same angles and proportional sides; congruent shapes are exactly the same size as well as shape.
Why does area scale by the square of the scale factor?+
Area depends on two lengths, so both are multiplied by the scale factor, giving the factor squared.
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