Similarity & congruence

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.

🔑

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.

📐

Formulas

Area scale factor

ASF = k²

k is the linear scale factor

Volume scale factor

VSF = k³

✍️

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

⚠️

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.

🎯

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.

Ready to test yourself on Similarity & congruence?

Get AI-marked practice questions on exactly this subtopic.

Practice this topic →

← All topics Dashboard

▶️ Watch on YouTube

Free video lessons

Click a topic to search

Opens YouTube — pick any free GCSE video.