Area scales by the square of the scale factor

Double every length and the area does not double — it quadruples.

Every length in a similar figure scales by the same one number. Area does not.

Two triangles, ABC and A′B′C′. A live reading gives each triangle’s own area, the scale factor between AB and A′B′, and checks the area ratio against that scale factor squared.

Two triangles, ABC and A'B'C', all six points draggable and independent of each other. A live reading gives each triangle's area, the scale factor between AB and A'B', and checks the area ratio against the square of that scale factor. This needs JavaScript switched on.

Drag any point. The area ratio always matches the scale factor, squared.

Try to find a similar pair where the area ratio does not land on the scale factor squared. There isn’t one.

When one figure’s dimensions are $k$ times another’s, $k$ is their linear scale factor — “linear” specifically to separate it from what happens to area, which is not linear at all.

Why area outruns length

A triangle’s area is base times height, divided by two. In a similar triangle scaled by $k$ , the base is $k$ times as long and the height is $k$ times as tall — both are lengths, and every length scales by the same one factor. So:

A_{\text{scaled}} = \frac{(k \times \text{base}) \times (k \times \text{height})}{2} = k^2 \times \frac{\text{base} \times \text{height}}{2} = k^2 \times A_{\text{original}}

Both the base and the height contributed a factor of $k$ , and multiplying them together makes those two factors multiply too — $k \times k = k^2$ , not $2k$ . The same argument works for any shape that can be built from triangles, which is every polygon there is.

It’s tempting to assume “twice as big” means “twice the area” — doubling a photo’s dimensions looks like a modest change, but it actually needs four times the paper. That’s why enlarging a room’s floor plan or a printed image by a given percentage costs far more material than the percentage suggests.

One to watch out for.

This is the same idea, one dimension further, that a length scales by $k$ and a volume — three lengths multiplied together — would scale by $k^3$ . Area sits in between: two lengths, squared.

A photo is enlarged with a linear scale factor of 3. The original has area 20 cm². What is the enlarged area?

$20 \times 3^2 = 20 \times 9 = 180$ cm².

Two similar fields have a linear scale factor of 4 between them. The smaller field is 50 m². What is the larger field’s area?

$50 \times 4^2 = 50 \times 16 = 800$ m².

A map has scale 1:100. A park measures 3 cm² on the map. How large is the real park, in m²?

The linear scale factor is 100, so the area scale factor is $100^2 = 10\,000$ . The real park is $3 \times 10\,000 = 30\,000$ cm², which is 3 m² — remembering to convert the units only once, at the end.

Do the exercise

This lesson counts as done once you get its exercise right. Nothing to tick off by hand.