Spherical cap area upper bound
= Spherical cap area upper bound
{title2=$\operatorname{Vol}_{n-1}(D_\rho)\leq s_{n-1}\sin^n\rho$}
On the <unit sphere> $S^{n-1}$, let $D_\rho$ be a <spherical cap> of angular radius $0<\rho<\pi/2$. If $\sin\rho\geq1/\sqrt n$, its <surface area> is at most $s_{n-1}\sin^n\rho$. This includes $\rho\geq\pi/4$ for every $n\geq2$. The bound follows by comparing the cap-area derivative with the derivative of $\sin^n\rho$ and checking the two endpoint values.