 Solution

The preceding parts show that an area maximizer among curves of fixed length is a plane section of the sphere, hence a circle. If its smaller cap has angular radius $0\le \theta \le \pi /2$, thenFor this circle,No curve of the same length encloses more area than the maximizing circle. Equivalently every curve enclosing the smaller area $A$ satisfies the spherical isoperimetric inequalitywith equality precisely for circles.