Two circles of equal size intersect and the centre of each circle
is on the circumference of the other. What is the area of the
intersection? Now imagine that the diagram represents two spheres
of equal volume with the centre of each sphere on the surface of
the other. What is the volume of intersection?

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.

Spherical Triangles on Very Big Spheres

Age 16 to 18 Challenge Level:

Use the fact that for small values of x $$\cos(x) \approx 1 -
\frac{x^2}{2} + \frac{x^4}{24}$$

to find an approximation to the identity which holds for large
values of R.