Determination of areas on the plane, sphere and ellipsoid

Abstract This paper shortly reviews various methods to determine the area of a closed polygon on the plane, sphere and ellipsoid. A new method is derived for calculating the area of a geodetic polygon, i.e. a polygon on the ellipsoid limited by sections of geodesics. By a recursive procedure the area can be determined to any desired order of the eccentricity of the ellipsoid. Finally we also present a direct method for numerical integration of the area under the geodesic.