Scale factors and scale diagrams

Edexcel Foundation routinely tests map and scale-drawing reading. Higher uses similar figures (linked to G19) where scale factors of length, area, and volume relate via SF, SF², SF³.

Scale notation

A scale of 1 : 50 000 means 1 unit on the map = 50 000 of the same units in real life. 1 cm on the map = 50 000 cm = 500 m = 0.5 km.

A common UK Ordnance Survey scale: 1 : 25 000 (popular with Edexcel) means 1 cm = 250 m = 0.25 km.

Reading a map (real distance from scale)

real distance = map distance × scale ratio (in same units, then convert).

Example: distance on a 1:50 000 map is 8 cm. Real distance = 8 × 50 000 = 400 000 cm = 4 km.

Drawing a scale diagram

map distance = real distance ÷ scale ratio (after converting to same units).

Example: a field 200 m long on a 1:5000 scale drawing. 200 m = 20 000 cm. 20 000 ÷ 5000 = 4 cm on the drawing.

Scale factors of similar shapes

If two shapes are similar with linear scale factor k:

• Lengths scale by k.
• Areas scale by k².
• Volumes scale by k³.

Example: two cones similar with SF 2 (the larger has twice the radius).

• Volume ratio = 2³ = 8.
• Area ratio = 2² = 4.

Edexcel exam tip

For map work, always convert units before doing arithmetic. Going from cm to km: ÷ 100 000 (cm to m is ÷100, m to km is ÷1000).

For scale-factor questions on Higher, watch for the direction:

• "How many times bigger is the larger volume?" — use SF³.
• "What is the linear scale factor between the volumes?" — find the cube root.

Common Edexcel question pattern

A scale drawing shows a garden at a scale of 1 : 200. The garden is 6 cm long on the drawing. What is the actual length in metres?

Solution: 6 × 200 = 1200 cm = 12 m.