SPHERICAL TRIANGLE
The theodolite measures horizontal angles in the horizontal plane, but when the area becomes large, such as in the case of primary triangulation, the curvature of the earth means that such planes in large triangles called as spherical triangles or geodetic triangles are not parallel at the apices as shown in Fig. 6.8. Accordingly, the three angles of a large triangle do not total 180°, as in the case of plane triangles, but to 180° + £, where £ is known as spherical excess. The spherical excess depends upon the area of the triangle, and it is given by
£= Ao / R sin1’ seconds
where A = the area of the triangle in sq km, and
R = the mean radius of the earth in km (=6373 km).
The triangular error is given by
£ = Σ Observed angles – (180° + £)
= A + B + C – (180° – £)