Spherical isoperimetric inequality
= Spherical isoperimetric inequality
A simple closed curve of length $L$ on the unit sphere enclosing the smaller area $A$ satisfies
$$
L^2\geq A(4\pi-A).
$$
Equality holds for a circle. A variational proof uses Gauss--Bonnet to turn area maximization at fixed length into geodesic-curvature minimization, proves that an extremizer has constant geodesic curvature and is planar, and evaluates its spherical cap.