isepiphanic inequality

The classical isepiphanic inequality

$$36\pi V^2\leqq A^3,$$

concerns the volume $V$ and the area $A$ of any solid in ${ℝ}^3$.  It asserts that the ball has the greatest volume among the solids having a given area.

For a ball with radius $r$, we have

$$V=\frac{4}{3}\pi r^3,A=\mathrm{\hspace{0.33em}4}\pi r^2,$$

whence it follows the equality

$$36\pi V^2=A^3.$$

Cf. the isoperimetric inequality and the isoperimetric problem.