Solid angle on a sphere of the intersection of multiple hemispheres

Question

Consider the following scenario where I have a unit sphere cut by $N$ halfplanes that all contain the center of the sphere, thus forming several hemispheres. How can I calculate the solid angle subtended by the intersection of those hemispheres?

The problem is trivial when $N=2$, but what about arbitrary $N$ (I have a practical problem that requires me to solve for the case when $N=4$)? The intersection area is a spherical convex polygon(is it?), so is there a closed-form solution like calculating the area of a Euclidean 2D convex polygon?

Answer 1

From wikipedia:

Consider an N-sided spherical polygon and let $A_n$ denote the $n$-th interior angle. The area of such a polygon is given by (Todhunter,1 Art.99)

$${\text{Area of polygon (on the unit sphere)}}\equiv E_{N}=\left(\sum_{n=1}^{N}A_{n}\right)-(N-2)\pi .$$