gen-define-spherical-geometry-and-give-one-short-worked-example-3c75

Define spherical geometry and give one short worked example.

# Spherical Geometry **Spherical geometry** is the geometry of the surface of a sphere, where "lines" are great circles (circles whose centers coincide with the sphere's center), and the usual rules of Euclidean geometry don't all apply. Key differences from plane geometry: - The sum of angles in a triangle *exceeds* 180° - There are no parallel lines (all great circles intersect) - The ratio of circumference to diameter is *less than* π ## Worked Example: Triangle Angle Sum **Problem:** Find the sum of angles in a spherical triangle formed by the equator and two meridians (lines of longitude) that meet at the North Pole at a 90° angle. **Solution:** The triangle has vertices at: - North Pole (N) - Two points on the equator The two meridians meet at right angles at the pole, so: - Angle at N = 90° Each meridian meets the equator at a right angle (this is a property of meridians), so: - Both angles at the equator = 90° each **Total angle sum = 90° + 90° + 90° = 270°** This exceeds 180° by 90°, demonstrating that spherical triangles violate the Euclidean angle sum rule. (The excess angle is actually related to the triangle's area!)