Find a closed plane curve of a given perimeter which encloses the greatest area. The solution is a circle.
If the class of curves to be considered is limited to smooth curves, the isoperimetric
problem can be stated symbolically as follows: find an arc with parametric
equations ,
for
such that ,
(where no further intersections occur) constrained by
Zenodorus proved that the area of the circle is larger than that of any polygon having the same perimeter, but the problem was not rigorously solved until
Steiner published several proofs in 1841 (Wells 1991).