Area of a circle on sphere

If $S$ is a sphere of radius $R$ in Euclidean three-space, $p$ is a point of $S$, and $S'$ is a sphere of radius $r$, $0 < r \leq 2R$, centered at $p$, then the area of the portion of $S$ inside $S'$ is $\pi r^{2}$. (!)

(A detailed argument is given in my answer to Why do disks on planes grow more quickly with radius than disks on spheres?. I learned this nifty fact way back from Vector Calculus by Marsden and Tromba.)

If instead the "radius" $0 < r \leq \pi R$ of a disk on a sphere of radius $R$ is measured intrinsically (as an arc length of a great circle along the sphere), the area is $$ 2\pi R^{2}\left(1 - \cos\frac{r}{R}\right), $$ as can be shown by straightforward calculation.