Spherical Pythagorean Theorem

Did you know there is a version of the Pythagorean Theorem for right triangles on spheres? First, let's define precisely what we mean by a spherical triangle. A great circle on a sphere is any circle whose center coincides with the center of the sphere. A spherical triangle is any 3-sided region enclosed by sides that are arcs of great circles. If one of the corner angles is a right angle, the triangle is a spherical right triangle. In such a triangle, let C denote the length of the side opposite right angle. Let A and B denote the lengths of the other two sides. Let R denote the radius of the sphere. Then the following particularly nice formula holds:   cos(C/R) = cos(A/R) cos(B/R). Presentation Suggestions: Verify the formula is true in some simple examples: such a triangle with two right angles formed by the equator and two longitudes. For more on spherical triangles, see the Fun Fact on Spherical Geometry. The Math Behind the Fact: This formula is called the "Spherical Pythagorean Theorem" because the regular Pythagorean theorem can be obtained as a special case: as R goes to infinity, expanding the cosines using their Taylor series and manipulating the resulting expression will yield:   C2 = A2 + B2 as R goes to infinity! This should make sense, since as R goes to infinity, spherical geometry becomes more and more like regular planar geometry!