Legendre's theorem on spherical triangles

Legendre's theorem states that subtracting one-third of a small spherical triangle’s excess from each angle gives a plane triangle whose angles and sides approximate the spherical triangle.

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Spherical excess: The amount by which a spherical triangle’s angle sum exceeds 180 degrees. Its one-third is subtracted from each spherical angle in the theorem.
Geodesy: The science of measuring Earth’s shape, orientation, and gravitational field. Classical geodetic computations use Legendre’s approximation for small Earth-surface triangles.
Euclidean geometry: The geometry of flat space governed by Euclid’s postulates. The theorem produces a Euclidean triangle as an approximation to a spherical one.
Spherical law of cosines: A formula relating the sides and an angle of a spherical triangle. It solves spherical triangles directly rather than replacing them with corrected planar ones.
Spherical triangle: A triangle formed by three arcs of great circles on a sphere. The theorem starts with this curved figure and approximates it in a plane.
Triangulation: A surveying method that locates points by measuring angles in connected triangles. Small spherical survey triangles can be reduced to plane triangles using the theorem.
Triangle angle sum: The sum of a triangle’s interior angles; it is 180 degrees in Euclidean geometry. The theorem adjusts spherical angles toward this planar angle sum.
Spherical law of sines: A relation stating that each spherical side’s sine is proportional to the sine of its opposite angle. It offers a direct spherical-trigonometry method without Legendre’s planar reduction.
Great-circle distance: The length of the shortest path between two points on a sphere, measured along a great circle. These spherical side lengths are approximated by the corrected plane triangle’s sides.
Geodetic datum: A reference system defining coordinates and the Earth model used in geodetic measurements. Datum-based ground measurements may require spherical or ellipsoidal triangle calculations.