Spherical Excess

On a 2 dimensional surface a triangle’s interior angles always sum to 180°. On the surface of a sphere a triangle’s interior angles sum to over 180° and less than 540°. See here for more details on the math on spherical triangles: https://mathworld.wolfram.com/SphericalTriangle.html

Long range surveys across large areas measure thousands of triangles as a result of the survey work. These triangles always sum to more than 180 degrees. See links below for a list of publications with examples of measured spherical triangles.

Knowing all the angles of a spherical triangle and one or more of the lengths of the sides it is possible to determine the radius of the sphere upon which the triangle sits. The formula is cos(c/R) = (cos(C) + cos(A)*cos(B)) / (sin(A)*sin(B)). Where R is the radius of the sphere, A, B, and C are interior angles and c is the length of the side opposite angle C.

Solving for R we get R = c / (acos( csc(A) * csc(B) * cos(C) + cot(A) * cot(B))). See screenshot from WolframAlpha for this solution.

This formula can identify the radius of any spherical object just by knowing the dimensions of a spherical triangle on its surface.

I have put this formula into a spreadsheet, linked below, and entered the measurements from several spherical triangles. This method uses zero assumptions to measure the radius of the earth.

Link to spreadsheet: https://docs.google.com/spreadsheets/d/1kus6gZDIdR_Q3W3OnW0hNyn35CUWas5szyz_dWRwj0Q/edit?usp=sharing

Since the formula only uses one side I have arranged the spreadsheet to calculate the radius using all three sides and average the values. This spreadsheet also calculates the mean of all the different triangles.

Surveys with spherical triangles:


Illustration of spherical triangles vs planar triangles on the surface of a sphere.