spherical excess

Since the area of a triangle cannot be negative the spherical excess is always positive.

The spherical excess is proportional to the area of the triangle.

It was the first instrument to be able to measure the spherical excess of large survey triangles.

In geodetic operations the observations are adjusted for spherical excess and projection variations.

In practical applications it is often small: for example the triangles of geodetic survey typically have a spherical excess much less than 1' of arc.

This follows from the theory of spherical excess and it leads to the fact that there is an analogous theorem to the theorem that "The sum of internal angles of a planar triangle is equal to ", for the sum of the four internal solid angles of a tetrahedron as follows: