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

A triangle on a flat surface always sums to 180 degrees. But draw one on a sphere, and you can get 270 degrees — three perfect right angles.

The Gauss-Bonnet theorem explains it all: the angular excess equals the Gaussian curvature integrated over the area of the triangle.

Positive curvature: angles exceed 180 degrees.
Negative curvature: angles fall below 180 degrees.
Zero curvature: exactly 180 degrees.

And since Einstein showed that mass curves spacetime, no physical triangle in our universe truly sums to exactly 180 degrees.

Math Pitfalls — where the rules you learned in school break down.

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