Three points are randomly chosen on the surface of a sphere. They are connected to each other by great arcs to form a (curved) triangle. What is the expected value of the area of the triangle?
We know from Girard's theorem that the area of a spherical triangle is:
$ A= R^2 E $
Where R is the radius of the sphere and E is the excess angle of (α+β+γ−π). (α, β and γ are the angles of the triangle.)
However, I'm struggling to work out how to calculate the expected area of the triangle.
Note: An analytic solution would be preferable, but a brute-force solution would also be ok.
