Recently my differential geometry lecturer demonstrated that the sum of the interior angles of a triangle in a sphere is not necessarily never $180^\circ$. This is one way to prove that the earth is not flat. I was wondering, what then is the maximum sum of the interior angles of triangles in a sphere, since this sum is not a constant?
Geometry – Maximum Sum of Angles in a Spherical Triangle
differential-geometrygeometryspherical-geometry
Best Answer
Perhaps your teacher taught you something like this from Wikipedia: $$180^{\circ}\times\left(1+4 \tfrac{\text{Area of triangle}}{\text{Surface area of the sphere}}\right)$$
If you are prepared to have a triangle which has more than half the area of the sphere then the maximum can approach $900^\circ$; if not then $540^\circ$.