Is there a syntax to use with MMA Integrate-Over-Regions to calculate the area inside a "polygon" projected onto the surface of a sphere?

Background. I have a polyhedron and I'm looking for the solid angle of a vertex. I can put a sphere around that vertex. I can project the ordered list of edges incident on that vertex to an ordered list of points making up a "polygon" on the surface of the sphere. I'm looking for a simple way to calculate the area inside that "polygon" on the sphere. It will be convex if that matters. I've guessed various different ways that this might be expressed and Integrate over regions rejected all of them.

Thanks