Areas of Similar Figures

Area scales by the square of the similarity ratio.

The idea

Two figures are similar when one is a scaled copy of the other: the same shape, possibly a different size. The similarity ratio $k$ is the number every length is multiplied by in passing from the first figure to the second. It applies to every matching length at once — sides, heights, diagonals, radii — so one pair of corresponding measurements determines it.

Theorem (Areas of similar figures).

If two figures are similar with similarity ratio $k$, so that every length of the second is $k$ times the matching length of the first, then the area of the second is $k^{2}$ times the area of the first.

Lengths scale by $k$, but areas scale by $k^{2}$. Double every length of a figure ($k = 2$) and the enlarged figure holds four copies of the original, not two.

To pass from a length ratio to an area ratio, square it; to recover a length ratio from an area ratio, take a square root. Areas in the ratio $100 : 1$ mean lengths in the ratio $10 : 1$.

Ways to work on it

• Walkthrough. Length ratio squared gives the area ratio.
• Proof. Why area scales by k^2: tile with unit squares and scale each.
• Practice. Given a length ratio and one area, find the other.
• Hardest. Go backwards: area ratio to length ratio via a square root.

Not sure where to start? Take the ten-question placement test.