Geodesic Polygon Area Calculator
The area of a triangle drawn on the curved surface of the Earth is larger than the flat-map area, because the sum of its angles exceeds 180 degrees. That surplus, the spherical excess, multiplied by the square of the Earth radius gives the true geodesic area (Girard's theorem). This calculator takes the latitude and longitude of three vertices, computes the three great-circle side lengths, applies the spherical law of cosines for the angles, and returns the geodesic area in square metres, hectares and acres on the WGS84 mean radius.
Geodesic triangle area formula
a, b, c = great-circle side lengths (angular, radians)
angle A = acos((cos a - cos b cos c) / (sin b sin c))
similarly for B and C
excess E = A + B + C - pi
area = E * R^2
Sides are the central angles between vertices (great-circle distance over R). The spherical law of cosines gives each interior angle. Girard's theorem turns the angle excess into area. R defaults to the WGS84 mean radius.
Geodesic area context
- Geodesic area accounts for Earth curvature; flat-map area underestimates large regions.
- Spherical excess is the surplus over 180 degrees in the triangle's angle sum.
- Girard's theorem: area equals excess (radians) times radius squared.
- The WGS84 mean radius (6,371,008.8 m) gives sub-one-percent error for most triangles.
- Split many-sided polygons into triangles and sum their geodesic areas.
Geodesic area: frequently asked questions
How is geodesic area different from planar area?
Planar area assumes a flat surface, which underestimates large areas on the curved Earth. Geodesic area accounts for the curvature. For a spherical triangle, the area equals the spherical excess (the amount its angles exceed 180 degrees) times the radius squared.
What is spherical excess?
On a sphere the interior angles of a triangle sum to more than 180 degrees. The excess E (in radians) is that surplus. By Girard's theorem, the triangle's area is E times R squared, where R is the sphere radius. A larger triangle has a larger excess.
Which radius does this use?
It uses the WGS84 mean Earth radius of 6,371,008.8 metres by default, and you can change it. The Earth is an oblate spheroid, so a single mean radius gives a close spherical approximation; for survey-grade ellipsoidal areas use a dedicated geodesic library.
How accurate is the spherical approximation?
Treating the Earth as a sphere of mean radius introduces an error of well under one percent for most regions because of the small flattening (1/298.26). For modest triangles the result is close to the true ellipsoidal geodesic area; very large or high-precision areas need an ellipsoidal method.
Can I use this for a parcel of more than three corners?
This page handles a single spherical triangle. For a polygon with more vertices, split it into triangles from one vertex, compute each triangle's geodesic area here, and sum them. A planar shoelace method on a projected grid is an alternative for small parcels.
Official sources
- NIST Digital Library of Mathematical Functions: spherical trigonometry reference.
- NOAA National Geodetic Survey: geodetic datums and the WGS84 ellipsoid.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.