Length, area and volume scale factors

When every length is multiplied by \(k\), two perpendicular dimensions make area multiply by \(k^2\); three dimensions make volume multiply by \(k^3\). The figures or solids must be similar for these general rules.

Example

Two similar solids have volume ratio \(27:125\). Their length ratio is \(3:5\), so their surface-area ratio is \(9:25\). If the smaller surface area is \(72\text{ cm}^2\), the larger is \(72\times\dfrac{25}{9}=200\text{ cm}^2\).

Triangles with the same perpendicular height instead have area ratio equal to their base ratio, even if they are not similar: \(\dfrac{\tfrac12 b_1h}{\tfrac12 b_2h}=\dfrac{b_1}{b_2}\). Do not square a base ratio unless similarity is established. Masses follow volume ratios only when densities are equal.