Spherical Excess Calculator

About this calculator

On a flat plane the angles of a triangle always sum to 180°, but on the surface of a sphere they sum to more than 180° — the difference, called the spherical excess E = A + B + C − 180°, grows with the size of the triangle relative to the sphere. Girard's theorem ties this excess directly to area: the triangle's surface area equals E (expressed in radians) times the square of the sphere's radius, Area = E·R².

This relationship underlies classical geodesy and celestial navigation, where surveyors and navigators working with large-scale distances on Earth's curved surface cannot use flat-plane trigonometry without introducing error. It is also the basis for computing areas of large regions bounded by great-circle arcs, such as in astronomy for spherical polygons on the celestial sphere.