Among quadrilaterals with sides of specified lengths, there is a unique quadrilateral that encloses the maximum possible area. This extremal quadrilateral is cyclic, that is, its vertices lie on a circle. The extremal pentagons are also cyclic. In general, n-sided polygons with sides of specified lengths that enclose the maximum area are cyclic. These facts are used to solve the isoperimetric problem: Among closed plane curves of length $L$, find the curves that enclose the maximum area. Solution: the circles of length $L$. \vskip 3in