What is the minimum and maximum of sum of angles of a spherical triangle? Let us remove a constraint from spherical triangles: sides are not necessarily circular arcs. Then what will be the minimum and maximum of sum of angles of such triangles?
[Math] Sum of angles of a triangle on a sphere
general-topologygeometryspherical-geometry
Best Answer
A triangle with zero interior angle sum:
(The horizontal line is not part of the triangle. It's the line containing the diameters of the semicircles that are the edges of the triangle. $IJK$ is a triangle with straight edges in a hyperbolic space.)
Although that triangle is drawn on the plane, it should be no great challenge to use it as a model to produce a triangle on the sphere with $0$ interior angle sum. It's exterior is also a figure bounded by three sides and three vertices, so is also a triangle and the angle sum of its exterior is clearly $6\pi$.