Area in d dimensions of a spherical part

Question

Let $S^{d-1}$ be the unit sphere (centered at $O$) in $d$ dimensions. One can show that when $d=3$, for fixed $x\in S^{2}$ the area of $P(x) = \{y\in S^{2}, \angle xOy < \alpha \}$ is $2\pi(1-\cos\alpha)$. What is the analogous result in $d$ dimensions? Any help appreciated.