Four generalizations of the Pythagorean theorem

Here are four theorems that generalize the Pythagorean theorem. Follow the links for more details regarding each equation.

1. Theorem by Apollonius for general triangles.

2. Edsgar Dijkstra’s extension of the Pythagorean theorem for general triangles.

$\text{sgn}(\alpha + \beta - \gamma) = \text{sgn}(a^2 + b^2 - c^2)$

3. A generalization of the Pythagorean theorem to tetrahedra.

$V_0^2 = \sum_{i=1}^n V_i^2$

4. A unified Pythagorean theorem that covers spherical, plane, and hyperbolic geometry.

$A(c) = A(a) + A(b) - \kappa \frac{A(a) \, A(b)}{2\pi}$

One thought on “Four generalizations of the Pythagorean theorem”

Brian Oxley

14 November 2025 at 06:57

What a nice post, everything in one place. How does it work for steradia, that is, tetrahedra from a sphere’s center to a triangle on it’s surface? Does the sphere need to be a sphere, or can the surface triangle be over any closed continuous solid when the tetrahedra do not cross the surface twice?

I wish I had taken diffy g at uni!

One thought on “Four generalizations of the Pythagorean theorem”

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