# Dictionary:Spherical excess

The amount by which the sum of the angles of a triangle on the surface of a sphere exceeds 180 degrees. For a sphere, this excess is the area of the triangle. For a spherical triangle on an ellipsoid, such as the Earth approximates, the spherical excess ${\displaystyle \varepsilon }$ is approximately:
${\displaystyle \varepsilon =mbc{\text{sin}}\alpha }$,
where b and c are two adjacent sides of the triangle which intersect at the angle ${\displaystyle \alpha }$, m = latitude function=${\displaystyle {\frac {\rho }{2RN}}}$, ${\displaystyle \rho }$=number of seconds of arc/radian=206 264.8, R=radius of curvature in the meridian, and N=radius of curvature in the prime vertical. Values of m for various ellipsoids are obtained from tables; the value for the center of the triangle to the nearest half degree is usually used. For greater precision a correction factor is often applied in iterative fashion.