At Higher tier, similarity is extended to area and volume scale factors. Congruence requires proving triangles identical using the four conditions: SSS, SAS, ASA and RHS.
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Key facts to remember
1If linear scale factor is k, then area scale factor is k², and volume scale factor is k³.
2Two triangles are congruent if they satisfy: SSS, SAS, ASA (or AAS) or RHS.
3Similar triangles have equal corresponding angles and sides in the same ratio.
4To find the area of a similar shape: Area₂ = Area₁ × k².
5To find the volume of a similar solid: Volume₂ = Volume₁ × k³.
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Formulas
Area scale factor
ASF = k²
k is the linear scale factor
Volume scale factor
VSF = k³
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Worked examples
Example 1
Two similar cylinders have heights 4 cm and 10 cm. If the smaller has volume 48 cm³, find the volume of the larger.
Working
Linear scale factor k = 10 ÷ 4 = 2.5
Volume scale factor = k³ = 2.5³ = 15.625
Larger volume = 48 × 15.625 = 750 cm³
Answer750 cm³
Example 2
Two similar triangles have areas 9 cm² and 25 cm². A side in the smaller triangle is 6 cm. Find the corresponding side in the larger triangle.
Working
Area scale factor = 25/9
Linear scale factor = √(25/9) = 5/3
Corresponding side = 6 × (5/3) = 10 cm
Answer10 cm
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Common mistakes
✗Using the linear scale factor for area (should be k²) or volume (should be k³).
✗Forgetting to square root the area ratio to find the linear scale factor.
✗Confusing congruence conditions: RHS (right angle, hypotenuse, side) is not the same as SSS.
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Exam tips
✓Always identify k (linear scale factor) first, then square for area or cube for volume.
✓For congruence proofs, state the condition used (SSS, SAS, ASA or RHS) explicitly.
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