Geodesic Triangle: Calculating Area on a Sphere

[dori1123]

An equilateral geodesic triangle is right angled. The area of a geodesic triangle on a sphere of radius r is (1/2)$$\pi$$r^2. But how is that obtained?

 

[adriank]

Draw one on a sphere, and it'll cover 1/8 of the surface.

Or you could use Girard's theorem: the area of a spherical triangle is r²e, where e is the excess angle of the triangle (the sum of the angles, minus pi).