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 3sided 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.
<http://www.math.hmc.edu/funfacts>.
