[ants_everywhere]

If you run Buffon's needle a lot you come to the conclusion that we live in a locally Euclidean space. That's fine and good. We also live in what you might consider a "locally Newtonian" world, but when things get very big or very small, the Newtonian approximation breaks down.

The ratio of a circle's circumference to its diameter (or equivalently the sum of triangle angles) has the same problem. If you want general relativity to work, then we need to live in a curved spacetime. Depending on whether that spacetime is positively or negatively curved, the angles of a very large triangle may add up to more than or less than pi radians.