EQN-02-04

Scale, Maps & Similar Figures

Use a scale factor to move between a drawing and reality, and between two similar shapes.

~5 min read•Difficulty 2/5

Scale as a ratio

A scale "1 : 50 000" means 1 unit on the map equals 50 000 of the same unit in reality. Set up a proportion and cross-multiply.

map : real = 1 : 50000

3 cm on the map \to 3 \times 50000 = 150000 cm = 1.5 km

Similar figures

Two shapes are similar when their angles match and their sides are in a constant ratio (the scale factor k).

Corresponding lengths scale by k.
Areas scale by k².
Volumes scale by k³.

If a model is built at scale 1 : 4 (k = 4) :

length \times 4, area \times 16, volume \times 64

The area/volume trap

Doubling the side of a square does not double its area — it quadruples it $(2^{2} = 4) .$ This single fact decides many geometry-flavoured ratio questions.

Worked examples

Example 1 (difficulty 2/5)

On a map with scale 1 : 200 000, two towns are 4 cm apart. What is the real distance?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Two similar rectangles have sides in the ratio 2 : 3. What is the ratio of their areas?

◎ Predict the answer first:

Tips

Write the scale as map : real, then cross-multiply — units stay straight.
Lengths scale by k, areas by k², volumes by k³ — memorize the powers.
Convert units at the very end, once the arithmetic is done.

Common mistakes

Scaling area by k instead of k²

Why: Area feels like "just another length".

Fix: Area is two-dimensional, so it scales by the square of the length ratio.

Forgetting the cm

\to

km conversion on map problems

Why: The big number looks like the final answer.

Fix: Finish by converting to the unit the answer choices use.

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