Isoperimetric Problem

Find a closed plane curve of a given length 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 $x=x(t)$, $y=y(t)$ for $t\in[t_1,t_2]$ such that $x(t_1)=x(t_2)$, $y(t_1)=y(t_2)$ (where no further intersections occur) constrained by

$$l=\int_{t_1}^{t_2} \sqrt{x'^2+y'^2}\,dt$$

such that

$$A={\textstyle{1\over 2}}\int_{t_1}^{t_2} (xy'-x'y)\,dt$$

is a Maximum.

See also Dido's Problem, Isovolume Problem

References

Bogomolny, A. ``Isoperimetric Theorem and Inequality.'' http://www.cut-the-knot.com/do_you_know/isoperimetric.html.

Isenberg, C. Appendix V in The Science of Soap Films and Soap Bubbles. New York: Dover, 1992.