Let $C$ be a pointed polyhedral cone in $\mathbb{R}^n$ and let $S^{n1}$ denote the unit sphere in $\mathbb{R}^n$. Given a description of the supporting hyperplanes of $C$ is there an algorithm for computing the spherical measure of $C \cap S^{n1}$? I suppose you could randomly generate points uniformly distributed on the unit sphere, and test each point to see if it is in $C$. Is there a better way?

This is in general as hard as computing the volume of an Euclidean polytope, but there are reductions for even dimensional polytopes to volumes of lowerdimensional things (of which there may, of course, be an exponential number). See http://www.math.ru.nl/~heckman/Heck_7.pdf (he mostly talks about the hyperbolic case, but the spherical case is identical). 


For very special classes of cones, there are combinatorial formulas related to these questions, e.g. 

