Area & Volume Scale Factors

Bigger lengths, much bigger areas

When you enlarge a shape by a length scale factor k, every side is multiplied by k. But the area does not just grow by k — it grows by k^2. And the volume of a solid grows by k^3. This is one of the most useful — and most misremembered — facts in school geometry.

The reason is simple once you picture it. A shape lives in two directions: it has a width and a height. Enlarging stretches each direction by k, so the area, which multiplies the two, stretches by k \times k = k^2. A solid lives in three directions — width, height and depth — so its volume stretches by k \times k \times k = k^3.

\text{length} \times k, \qquad \text{area} \times k^2, \qquad \text{volume} \times k^3.

Swap a small pizza for one with double the diameter and you might expect twice as much to eat. You get four times as much! Doubling the width is a scale factor of k = 2, and area scales by k^2 = 2^2 = 4. A "double-size" pizza is really a quadruple-size meal — which is exactly why the big one is such good value.

small pizza  →  width ×2 →  big pizza   area ×4

See it: count the little squares

Start with a single unit square — a square of area 1. Enlarge it by a scale factor k and you get a k \times k square. How many of the little unit squares tile it? Exactly k^2 of them — so the area is k^2. Step through the reveal, then press Refresh for a new scale factor.

The last caption whispers the 3-D version: if the square were a solid cube, an enlargement by k would hold k^3 little unit cubes. A \times 2 cube holds 2^3 = 8 of them; a \times 3 cube holds 3^3 = 27.

Using it forwards: length factor → area & volume factor

Given the length scale factor, square it for the area factor and cube it for the volume factor.

Worked example. A rectangle has area 5 cm². Enlarge it by k = 4 and its new area is 5 \times 4^2 = 5 \times 16 = 80 cm² — not 5 \times 4 = 20 cm². The lengths quadrupled, but the area went up 16 times.

Blow a balloon up to twice its width and it does not hold twice the air — it holds 2^3 = 8 times as much, because a balloon is a solid filling three directions at once. This is why a slightly bigger hot-air balloon can lift so much more, and why a small change in the size of a bubble is a big change in the breath it takes to blow it.

small balloon  →  width ×2 →  big balloon   air ×8

Using it backwards: area ratio → length factor

Two similar shapes are the same shape at different sizes, so their areas are in the ratio k^2 : 1. To go the other way — from an area ratio back to the length ratio — you take the square root. For solids, the volumes are in the ratio k^3 : 1, and you take the cube root.

k = \sqrt{\tfrac{\text{area}_{\text{big}}}{\text{area}_{\text{small}}}}, \qquad k = \sqrt[3]{\tfrac{\text{volume}_{\text{big}}}{\text{volume}_{\text{small}}}}.

Worked example. Two similar shapes have areas in the ratio 9 : 1. The length scale factor is \sqrt{9} = 3 — the bigger shape has sides 3 times as long, not 9 times.

Worked example. Two similar water tanks have volumes in the ratio 8 : 1. The length scale factor is \sqrt[3]{8} = 2, so the big tank is only twice as tall as the small one — even though it holds eight times the water.

The scale-factor traps that catch nearly everyone: