“Straight” depends on the surface
On a flat page, a triangle’s sides are straight lines, and its interior angles total 180 degrees. On a curved surface, the comparable paths are geodesics: routes that feel straight while staying on the surface. A globe’s meridians, for example, are geodesics even though they curve when drawn on a flat map.
How a globe makes a 270-degree triangle
Start at the equator and travel north along one longitude until you reach the North Pole. From there, head south along a longitude 90 degrees away, then follow the equator back to your starting point. Those three surface routes enclose a triangle.
Each corner is a right angle: the two meridians meet at 90 degrees at the pole, and each meridian meets the equator at 90 degrees. The total is therefore 270 degrees. The globe has not changed the meaning of an angle; its curved surface lets the triangle enclose more area than a flat triangle can.
Curvature changes the angle total
A saddle-shaped surface bends in opposite directions across its two dimensions. For a triangle whose sides follow geodesics, that negative curvature makes the interior angles add to less than 180 degrees. The difference from 180 depends on the curvature across the area enclosed—not on a failure of geometry, but on the shape of the space where the triangle is drawn.
So the familiar 180-degree rule is specific to flat geometry. On a curved surface, measuring a triangle’s angles can reveal the surface beneath it: a surplus points to sphere-like curvature, while a deficit points to saddle-like curvature.
