# Unified Pythagorean Theorem

A few days ago I wrote that the spherical counterpart of the Pythagorean theorem is

cos(c) = cos(a) cos(b)

where sides a and b are measured in radians. If we’re on a sphere of radius R and we measure the sides in terms of arc length rather than in radians, the formula becomes

cos(c/R) = cos(a/R) cos(b/R)

because an of length x has angular measure x/R.

How does this relate to the more familiar Pythagorean theorem on the plane? If a, b, and c are small relative to R, then the plane Pythagorean theorem holds approximately:

c² ≈ a² + b².

## Unified Pythagorean Theorem

In this post I’ll present a version of the Pythagorean theorem that holds exactly on the sphere and the plane, and on a pseudosphere (hyperbolic space) as well. This is the Unified Pythagorean Theorem .

A sphere of radius R has curvature 1/R². A plane has curvature 0. A hyperbolic plane can have curvature k for any negative value of k.

Let A(r) be the area of a circle of radius r as measured on a surface of curvature k. Here area and radius are measured intrinsic to the surface. Then the Unified Pythagorean Theorem says

A(c) = A(a) + A(b) – k A(a) A(b) / 2π.

## Plane

If k = 0, the final term on the right drops out, and we’re left with the ordinary Pythagorean theorem with both sides of the equation multiplied by π.

## Sphere

On a sphere of radius R, the area of a circle of radius r is

A(r) = 2πR²(1 – cos(r/R)).

Note that for small x,

1 – cos(x) ≈ x²/2,

and so A(r) ≈ πr² when Rr.  (Notation explained here.)

When you substitute the above definition for A in the unified theorem and plug in k = 1/R² you get

cos(c/R) = cos(a/R) cos(b/R)

as before.

## Pseudosphere

In a hyperbolic space of curvature k < 0, let R = 1/√(-k). Then the area of a circle of radius r is

A(r) = 2π(cosh(r/R) – 1)

As with the spherical case, this is approximately the plane area when Rr because

cosh(x) – 1 ≈ x²/2

for small x. Substituting the definition of A for hyperbolic space into the Universal Pythagorean Theorem reduces to

cosh(c/R) = cosh(a/R) cosh(b/R),

which is the hyperbolic analog of the Pythagorean theorem. Note that this is the spherical Pythagorean theorem with cosines replaced with hyperbolic cosines.

 Michael P. Hitchman. Geometry with an Introduction to Cosmic Topology. Theorem 7.4.7. Available here.