Solid angle created from irregular polygon (over a sphere)

Question

I have an $n$-polygon on a sphere ($n\geqslant3$). In this example the vertices are $C,D,E,F,G,H,I,J,K$. Which solid angle alpha generate this polygon respect origin of the sphere? For $C,D,E,F,G,H,I,J,K$ are know coordinate Cartesian or polar.

I am searching for a "closed solution" without integrals, and when is the solid angle generated from polygon $KJIHGFEDC$ being $β=4π-α$?