Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 163

Fig. 5.1 Spherical triangle

a ﾼ ∠ BOC b ﾼ ∠ AOC c ﾼ ∠ AOB :

, ,

Given are:

TAT 0 EBE 0 FCF 0 A ﾼ ∠ B ﾼ ∠ C ﾼ ∠ :

, ,

Hence, the arc length of any side of the spherical triangle equals its subtended
face angle of the trihedral angle; the angles of the spherical triangle are equal to the
corresponding dihedral angles of the trihedral angle.

5.1.2 Spherical Excess

Spherical excess is the amount by which the sum of the angles of a spherical
triangle exceeds the sum of the angles of a plane triangle, denoted by ε , namely:

180 :

ε ﾼ A þ B þ C ð5:1Þ

The computational formula of ε is given by:

S
R 2 ,

ε ﾼ ð 5 : 2 Þ

where S denotes the area of the spherical triangle and R is the radius of the sphere.