Scale Factors & Similar Shapes
Scale Factors & Similar Shapes
Two shapes are similar if one is an enlargement of the other — they have the same angles and their sides are in the same ratio. Congruent shapes are the same size and shape.
Scale Factors
The scale factor tells you how many times bigger (or smaller) one shape is compared to another.
Scale factor = new length ÷ original length
Worked Example: Triangle A has sides 3 cm, 4 cm, 5 cm. Triangle B has sides 7.5 cm, 10 cm, 12.5 cm. Find the scale factor.
- 7.5 ÷ 3 = 2.5, 10 ÷ 4 = 2.5, 12.5 ÷ 5 = 2.5
- Scale factor = 2.5
If the scale factor is less than 1, the shape gets smaller (this is still called an enlargement in maths).
Finding Missing Sides in Similar Shapes
Worked Example: Two similar rectangles. The first is 6 cm by 10 cm. The second has a width of 15 cm. Find its length.
- Scale factor = 15 ÷ 10 = 1.5
- Length = 6 × 1.5 = 9 cm
Similar Triangles
Two triangles are similar if:
- All three angles are equal (AA), or
- All three pairs of sides are in the same ratio (SSS similarity), or
- Two sides are in the same ratio and the included angle is equal (SAS similarity)
Worked Example: In a triangle, DE is parallel to BC. AD = 4 cm, DB = 6 cm, DE = 5 cm. Find BC.
- Triangles ADE and ABC are similar (parallel lines give equal angles)
- Scale factor = AB/AD = (4 + 6)/4 = 10/4 = 2.5
- BC = DE × 2.5 = 5 × 2.5 = 12.5 cm
Area Scale Factor (Higher)
If the linear scale factor is k, the area scale factor is k².
Worked Example: Two similar shapes have corresponding lengths of 3 cm and 12 cm. The area of the smaller shape is 20 cm². Find the area of the larger shape.
- Linear scale factor = 12 ÷ 3 = 4
- Area scale factor = 4² = 16
- Area of larger shape = 20 × 16 = 320 cm²
Volume Scale Factor (Higher)
If the linear scale factor is k, the volume scale factor is k³.
Worked Example: Two similar cylinders. The smaller has height 5 cm and volume 100 cm³. The larger has height 15 cm. Find its volume.
- Linear scale factor = 15 ÷ 5 = 3
- Volume scale factor = 3³ = 27
- Volume = 100 × 27 = 2700 cm³
Summary of Scale Factor Relationships
| Quantity | Scale Factor |
|---|---|
| Length, perimeter, height | k |
| Area, surface area | k² |
| Volume, mass, capacity | k³ |
Working backwards: If the area scale factor is 9, the linear scale factor is √9 = 3. If the volume scale factor is 64, the linear scale factor is ∛64 = 4.
Worked Example: Two similar solids have surface areas 50 cm² and 450 cm². The volume of the smaller is 80 cm³. Find the volume of the larger.
- Area scale factor = 450 ÷ 50 = 9
- Linear scale factor = √9 = 3
- Volume scale factor = 3³ = 27
- Volume = 80 × 27 = 2160 cm³
Map Scales
A scale of 1 : 25,000 means 1 cm on the map represents 25,000 cm (= 250 m) in real life.
Worked Example: On a 1 : 50,000 map, two towns are 8 cm apart. Find the real distance in km.
- Real distance = 8 × 50,000 = 400,000 cm = 4,000 m = 4 km
Exam Tips
- Area scale factor = (linear SF)² and volume SF = (linear SF)³ — do NOT mix these up
- When finding the scale factor, always divide the larger by the smaller (or new by original) consistently
- Similar shapes have equal angles — if angles differ, the shapes are NOT similar
- In map questions, convert to sensible units at the end (usually km or m)