Triangle Angles DON'T Sum to 180° on Curved Surfaces

Added: 2026-05-25

[00:00:00]A triangle on a flat surface always has angles summing to 180°.

[00:00:05]One of the first rules of geometry, but what if the surface is not flat? Take a sphere. Start at the North Pole, walk straight down to the equator, turn 90°, walk a quarter of the way around, turn 90° again, walk straight back to the pole. Three right angles. 90 + 90 + 90 = 270°.

[00:00:25]That extra 90° is called the angular excess. On a saddle-shaped surface, the opposite happens. The angles squeeze inward and the sum falls below 180. The Gauss-Bonnet theorem captures this perfectly. The angular excess equals the Gaussian curvature times the area. Zero curvature, 180° exactly. Positive curvature, more. Negative curvature, less. And here's the punchline. Einstein showed that mass curves spacetime itself. So, in our universe, no triangle truly sums to exactly 180°.