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.

See also

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