Gauss-Bonnet Formula for Polygons:
Show that if $G$ is a convex polygon in an open hemisphere or in a hyperbolic plane, then
$H(G)$ = $2p - \sum a_{i}$ = $\sum b_i - (n - 2)p$ = $ Area(G)\cdot K$,
where $\sum a_{i}$ is the sum of the exterior angles, $\sum b_i$ is the sum of the interior angles, $n$ is the number of sides, and $K$ is the Gaussian curvature. Also prove this on sphere.
I can divide the convex polygon into triangles and apply
$H(D) = 2p - (a_1 + a_2 + a_3) = (b_1 + b_2 + b_3) - p$
to triangle and add up the results ($H(G) = 2p - \sum a_i$).
I not sure what to do after, I was wondering if anyone can help me out?