Scale Drawings and Area

A map scale of 1 cm to 5 km means every centimetre on the map is 5 km in the world.

Lengths are easy

7 cm on that map is 7 × 5 = 35 km.

Area is not

Doubling every side of a rectangle makes the area four times bigger, not twice.

Why four

Both the length and the width doubled, and area multiplies them, so 2 × 2 = 4.

Area scales by the square of the factor

Double every length and the area becomes four times as big, not twice. Triple the lengths and the area is nine times. The area factor is the length factor squared, because area involves two dimensions and each is being scaled.

Seeing why

A 3 by 4 rectangle has area 12. Doubled, it becomes 6 by 8 with area 48 — four copies of the original would fit inside. Drawing the four copies inside the enlarged rectangle makes the squaring obvious rather than something to accept.

Volume scales by the cube

Double every edge of a box and its volume becomes eight times as large. Three dimensions, so the factor is cubed. This is why a scale model that looks twice as big weighs eight times as much, and why large animals are not simply scaled-up small ones.

The mistake this prevents

People routinely assume that doubling a map's scale doubles the land area shown, or that a pizza twice the diameter feeds twice as many. Both are wrong by a factor of two. Knowing that area squares is genuinely useful outside the classroom.