 Convex regular spherical polygon

A nondegenerate convex regular spherical $n$-gon contained in an open hemisphere has equal interior angles $\alpha$ in the displayed interval. The spherical excess formula gives area $n\alpha -(n-2)\pi >0$. Conversely, put equally spaced vertices at colatitude $t\in (0,\pi /2)$ around a pole. Dividing each central triangle in half gives $\cot (\alpha /2)=\cos t\tan (\pi /n)$ by the spherical law of cosines. As $t$ increases, the angle ranges continuously from the Euclidean limiting angle to $\pi$.