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