Scaling laws for similar figures

The ratio of similarity does not act in the same way on lengths, areas and volumes: each quantity scales with a different power of $k$.

Property — Scaling laws for similar figures

If two figures are similar with ratio of similarity $k$, then:

• all corresponding segments are in the ratio $k$;
• the areas are in the ratio $k^2$;
• the volumes (for similar solids) are in the ratio $k^3$.

$\frac{\ell'}{\ell} = k, \qquad \frac{A'}{A} = k^2, \qquad \frac{V'}{V} = k^3.$

The idea is that area depends on two dimensions and volume on three: doubling the lengths ($k=2$) quadruples the areas and multiplies the volumes by eight.

Links

Topics: Similarity
Concepts: Scaling laws · Ratio of similarity · Similarity
Methods: Similarity criteria · Transformations and homotheties
Skills: Using formulae