Spherical Triangle Angle Sum in Geodetic Surveying

Geodetic surveying deals with mapping large areas of the Earth, which is a curved surface. Therefore, it utilizes spherical geometry principles.

Plane vs. Spherical Triangles

In standard Euclidean (plane) geometry, the sum of the internal angles of any triangle is always exactly 180 degrees.

However, a spherical triangle is formed by the intersection of three great circles on the surface of a sphere. The rules for its angles differ:

• The sum of the internal angles of a spherical triangle is always greater than 180 degrees.
• There is no upper limit strictly less than 540 degrees, but typically, practical applications involve sums between 180 and 540 degrees.

Geodetic Application

Since geodetic surveying accounts for the Earth's curvature, calculations involving triangles on the Earth's surface treat them as spherical triangles.

The excess of the sum of the angles over 180 degrees (called the spherical excess) is directly related to the area of the spherical triangle.

Conclusion

Therefore, for a spherical triangle encountered in geodetic surveying, the sum of its internal angles must be greater than 180 degrees.