Ratios of the Perimeter, Area, and Volume of Similar Figures

Property If two similar figures have a scale factor of $\frac{a}{b}$, then the ratio of their perimeters is $\frac{a}{b}$, the ratio of their areas is $\frac{a^2}{b^2}$, and the ratio of their volumes is $\frac{a^3}{b^3}$.

Examples Two similar rectangles have a scale factor of $\frac{3}{5}$. The ratio of their perimeters is $\frac{3}{5}$, and the ratio of their areas is $(\frac{3}{5})^2 = \frac{9}{25}$. Two similar spheres have a radius ratio of $1:10$. The ratio of their surface areas is $1^2:10^2 = 1:100$, and the ratio of their volumes is $1^3:10^3 = 1:1000$.

Explanation Don't do extra math! If you already know the side ratio (the scale factor) is $\frac{a}{b}$, you've got a massive shortcut. The perimeter ratio is exactly the same, $\frac{a}{b}$. For the area ratio, you just square it: $\frac{a^2}{b^2}$. And for 3D volume, you cube it: $\frac{a^3}{b^3}$.