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 x = x(t), y = y(t) for t element [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 = integral_(t_1)^(t_2) sqrt(x^(, 2) + y^(, 2))d t such that A = 1/2 integral_(t_1)^(t_2)(x y' - x' y) d t is a maximum.

circle | Dido's problem | double bubble | isoperimetric quotient | isoperimetric theorem | isovolume problem | perimeter