Rule of thumb · MathematicsNº 75 / 96

Area scales with the square, volume with the cube

Double every dimension of a shape and its area quadruples while its volume grows eightfold — the exponent, not the shape, decides how fast size compounds.

Why it works

This is why small animals lose heat faster relative to their mass (surface area) than large ones, and why doubling a tank’s dimensions needs eight times the material to hold eight times the liquid, not twice.

Do it exactly

Estimate with the rule, then check it against the calculator that models it properly.

Open Geometry — 2D & 3D →

More in Mathematics

← PreviousSOH-CAH-TOA is one triangle, memorised three ways Next →The mean is not the middle when data is skewed

Browse all 96 rules of thumb →