Isoperimetric Inequalities

Isoperimetric Inequalities are inequalities concerning the area of a figure with a given perimeter. They were worked on extensively by Lagrange.

If a figure in a plane has area $A$ and perimeter $P$ then $\frac{4\pi A}{P^2} \leq 1$ . This means that given a perimeter $P$ for a plane figure, the circle has the largest area. Conversely, of all plane figures with area $A$ , the circle has the least perimeter.