How to calculate the area of ​​a polygon on a sphere?

Question

Mathematics

There is a set of points with 3 types of coordinates: spherical - degrees, radians, Cartesian for a sphere with a radius of 5 c.u.
It is necessary to calculate the area of ​​a polygon on a sphere using a tuple (array) of points, and do not calculate in case of intersection of lines.
I tried using formulas from geodetic sites, it was calculated incorrectly.

Answer 1

https://analytics.ncsu.edu/sesug/2016/SD-122_Final...

Answer 2

Integrate the given spherical surface and get the area.
PS
Twice)

Answer 3

If the polygon is convex, then it can be quite easily divided into triangles and summed up their areas.