Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 5E (Geometry)

On the unit sphere, the spherical excess formula gives the area of a spherical triangle asFor a sphere of radius $R$, the right side is multiplied by $R^2$.

Join an interior point of a convex spherical $n$-gon to its vertices by geodesics. The resulting $n$ spherical triangles contain all polygon angles and angles summing to $2\pi$ at the interior point. Adding their spherical excesses givesFor a regular polygon with interior angle $\alpha$, positivity of area requires $n\alpha >(n-2)\pi$, while strict convexity requires $\alpha <\pi$. Both limiting values can be approached by regular polygons, so