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:

C^{2} = A^{2} + B^{2}

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>.

**References:**

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

**Fun Fact suggested by:**

Francis Su