How do you express the spherical excess in terms of triangle measurements?

Spherical excess is a concept used in spherical trigonometry, particularly when dealing with spherical triangles, which are defined on the surface of a sphere. The spherical excess of a triangle is calculated quite differently from the planar triangle angles.

In the context of spherical triangles, the spherical excess, denoted typically as "E," is established by the sum of the angles of the triangle. Specifically, for spherical triangles, the sum of the angles exceeds 180 degrees. To express this excess, you take the total angle sum of the triangle and subtract 180 degrees, resulting in the additional degrees that reveal how much the sum exceeds this flat reference.

When considering the triangle measurements, the spherical excess can indeed be understood as the extra angular measurement beyond the flat geometry's expectation. This understanding is crucial for applications in fields such as geodesy and navigation, where accounting for the curvature of the Earth is vital. Therefore, to accurately express the spherical excess, it's done in terms of additional degrees that go beyond the flat triangle sum of angles.