The area formula for a spherical triangle is often stated as a consequence of the Gauss-Bonnet theorem: $$\int_MKdA+\int_{\partial M}k_gds=2\pi\chi(M)$$ The idea is that in a spherical triangle $T$ on the unit sphere, all of the geodesic curvature is contained in the vertices, so $$\int_TdA+\sum\alpha=2\pi$$ where we sum over the turning angles (exterior angles) $\alpha$ of the vertices. Rearranging and stating in terms of interior angles $\alpha'=\pi-\alpha$, $$A(T)=\pi-\sum \alpha'$$
I was wondering how this method could be generalized to higher dimensions with the Chern-Gauss-Bonnet (CGB) theorem. In particular I would like to derive the hypervolume formula for a spherical $2n$-simplex on $S^{2n}$ with the CGB theorem in terms of its interior solid angles. I believe the form of the CGB theorem we want is Theorem II in this paper by Allendoerfer and Weil, which is a generalization to Riemannian polyhedra. However, I don't have great knowledge of differential geometry so I can't exactly see how Theorem II should simplify in the case of a spherical simplex.