Similarity & Scale Factors
Similarity in geometry means shapes have identical angles but proportional sides, where the ratio between corresponding sides remains constant. A scale factor describes this proportion: if a triangle with side 6 cm is similar to one with side 18 cm, the scale factor is 3. This concept appears throughout GCSE mathematics, particularly in Year 9 when students apply similarity alongside trigonometric ratios.
Why it matters
Scale factors appear extensively in real-world applications, from architectural drawings to digital photography. A house plan drawn at 1:50 scale means 1 cm on paper represents 50 cm in reality — the scale factor is 50. Map-making relies on consistent scaling: Ordnance Survey maps use ratios like 1:25,000, where 4 cm represents 1 km. In manufacturing, parts must scale proportionally whilst maintaining structural integrity. Engineers use scale factors when designing models for wind tunnel testing, often scaling aircraft models down by factors of 10 or 20. Photography and digital imaging apply scaling algorithms that preserve image quality by maintaining proportional relationships between pixels. Areas scale by the square of the linear scale factor, which affects material costs in construction projects.
How to solve similarity & scale factors
Similarity — Scale Factor
- Similar shapes have the same angles but proportional sides.
- Scale factor = new length ÷ original length.
- Multiply all sides by the scale factor to find corresponding sides.
- Areas scale by (scale factor)².
Example: Scale factor 2: side 3 → 6, area ×4.
Worked examples
Two similar squares have sides 4 cm and 20 cm. What is the scale factor?
Answer: 5
- Divide the larger side by the smaller side → 204 = 5 — Scale factor = 20 ÷ 4 = 5.
Triangle A has sides 3, 4, 5. Triangle B is similar with scale factor 3. Find B's sides.
Answer: 9, 12, 15
- Multiply each side by the scale factor → 3×3=9, 4×3=12, 5×3=15 — Each side of B = corresponding side of A × 3.
Two similar rectangles: one is 5×11, the other is 15×?. Find the missing side.
Answer: 33
- Find the scale factor from known sides → 155 = 3 — Scale factor = 15 ÷ 5 = 3.
- Apply scale factor to the missing side → 11 × 3 = 33 — Missing side = 11 × 3 = 33.
Common mistakes
- Confusing scale factor with the actual measurements — writing that a 4 cm side scaled by factor 3 becomes 7 cm instead of 12 cm
- Adding the scale factor instead of multiplying — calculating 6 + 2 = 8 instead of 6 × 2 = 12 when scaling by factor 2
- Applying linear scale factor to areas without squaring — stating that area increases by factor 3 instead of factor 9 when linear scale factor is 3