In class today our professor showed us the general area formula for the area of a convex spherical polygon with radius 1, which is $\text{area}(\text{spherical polygon}(\theta_{1},...,\theta_{n}))=\left(\sum_{i=1}^{n} \theta_{i} - (n-2)\pi\right).(1)$
I am wondering if the formula holds for a concave spherical polygon with radius 1. I went on and tried to take all $\theta$ as the angles that is formed by the intersection of arcs of the spherical polygon, but instead of getting (1) formula I get $\text{area}(\text{spherical concave polygon}(\theta_{1},...,\theta_{n}))=\left(\sum_{i=1}^{n} \theta_{i} - (n/3)\pi\right).(2)$
But I really want to generalize the area of concave spherical polygon to the area of concave spherical polygon-- is there neat way to pick some particular angles instead of just taking every angle that is the intersection of two arcs so that the convex spherical polygon formula works on a concave spherical polygon? Some advice or references will be greatly appreciated.