**Isoperimetry**literally means "having an equal perimeter". In mathematics, isoperimetry is the general study of geometric figures having equal boundaries.

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2 Generalisations to Other Spaces 3 Extensions and Applications of the Problem 4 External links |

## The Isoperimetric Problem in the Plane

The theorem is usually stated in the form of an inequality that relates the perimeter and area of a closed curve in the plane. If *P* is the perimeter of the curve and *A* is the area of the region enclosed by the curve, then the inequality states that

4π*A* ≤ *P*^{2}

For the case of a circle of radius *r*, we have *A* = π*r*^{2} and *P* = 2π*r*, and substituting these into the inequality shows that the circle does indeed maximize the area among all curves of fixed perimeter. In fact, the circle is the only curve that maximizes the area.

There are dozens of proofs of this classic inequality. Several of these are discussed in the Treiberg paper below. In 1901, Hurwitz gave a purely analytic proof of the classical isoperimetric inequality based on Fourier series and Green's theorem.

## Generalisations to Other Spaces

The isoperimetric theorem generalises to higher dimensional spaces and non-Euclidean spaces.

## Extensions and Applications of the Problem

## External links