Calculus of Variations practice

The circle uniquely maximizes enclosed area among closed curves of perimeter L.

The isoperimetric inequality states $4\pi{}A \le L^{2}$, with equality iff the curve is a circle.

Using a Lagrange multiplier on the constrained problem, the Euler–Lagrange equation forces the extremal curve to have constant curvature.

Among closed curves of a fixed enclosed area A, the perimeter-minimizing curve is a square.