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!

By the way, there is a “hyperbolic geometry” version, too. Can you guess what it says? See the reference.

How to Cite this Page: 
Su, Francis E., et al. “Spherical Pythagorean Theorem.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

D. Velian, “The 2500-Year-Old Pythagorean Theorem”, Math. Mag., 73(2000), 259-272.

Fun Fact suggested by:
Francis Su

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