In non-Euclidean (spherical and hyperbolic) geometry, the angle sum of a triangle deviates from π radians, quantified as angular excess (spherical case, sum > π) or angular defect (hyperbolic case, sum < π); Girard's theorem formalizes this by showing that a spherical triangle's area is directly proportional to its angular excess and to the square of the sphere's radius. This connects to the theory of geodesics — locally shortest paths generalizing straight lines to curved surfaces — and to the distinction between intrinsic geometry (properties, such as curvature and geodesic triangle angle sums, invariant under bending without stretching) and extrinsic geometry (properties dependent on how a surface is embedded in ambient space), foundational concepts within differential geometry.
In non-Euclidean (spherical and hyperbolic) geometry, the angle sum of a triangle deviates from π radians, quantified as angular excess (spherical case, sum > π) or angular defect (hyperbolic case, s…