Spherical excess

Spherical excess is the amount by which a spherical triangle’s interior angles sum to more than 180 degrees. On a sphere, its value equals the triangle’s area divided by the square of the sphere’s radius.

Spherical triangle: A region on a sphere bounded by three arcs of great circles. Its three interior angles are the angles whose surplus defines spherical excess.
Interior angle: The angle formed inside a polygon by two adjacent sides meeting at a vertex. Spherical excess is computed from the three interior angles of a spherical triangle.
Spherical area: The surface area of a region lying on a sphere. Multiplying spherical excess by the radius squared gives a triangle’s area.
Hyperbolic geometry: A non-Euclidean geometry with constant negative curvature, in which triangle angle sums are less than 180 degrees. Its triangles have an angular deficit where spherical triangles have an excess.
Gaussian curvature: A measure of a surface’s intrinsic curvature at a point, defined as the product of its principal curvatures. For a sphere, constant positive curvature makes angular excess proportional to area.
Euclidean geometry: The geometry of flat space, governed by Euclid’s postulates and the parallel postulate. Its triangle angle sum of 180 degrees supplies the baseline for the excess.
Surveying: The practice of measuring and mapping positions, distances, and boundaries on or near Earth. Large-area triangulation must account for angular excess caused by Earth’s curvature.
Flat geometry: Geometry on a surface with zero intrinsic curvature, such as the Euclidean plane. Flat triangles have no angular excess because their angles sum to 180 degrees.
Girard's theorem: A theorem stating that a spherical triangle’s area equals its spherical excess multiplied by the square of the sphere’s radius. It states the direct formula connecting this angle surplus to area.
Riemannian geometry: The study of smooth spaces equipped with a metric that defines lengths and angles. It provides the broader framework for measuring geometry on curved surfaces.

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