Math CentralQuandaries & Queries


Question from Rohit, a student:

How do I find the angles of a triangle drawn on a sphere


We need more information for a complete answer. We first have to agree on what is meant by a triangle on a sphere:

I would define the vertices of the triangle to be three points on the sphere, and the sides of the triangle to be the curves drawn on the sphere that follow the shortest distance between the vertices.

Such curves are obtained by intersecting the surface of the sphere with the plane determined by the center of the sphere and the two vertices. (We call such curves "great circles".) The angle between two sides of a triangle is therefore the angle between the two planes (which is usually found by measuring the angle between the lines through the center of the sphere that are perpendicular to those planes and subtracting that number from 180 degrees). Your question then reduces to the problem of finding the angles between pairs of lines through the center of the sphere.


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