**Hyperbolic Geometry**

For a right triangle in hyperbolic geometry with sides *a*, *b*, *c* and with side *c* opposite a right angle, the relation between the sides takes the form:

where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles:

with γ the angle at the vertex opposite the side *c*.

By using the Maclaurin series for the hyperbolic cosine, cosh *x* ≈ 1 + *x*2/2, it can be shown that as a hyperbolic triangle becomes very small (that is, as *a*, *b*, and *c* all approach zero), the hyperbolic relation for a right triangle approaches the form of Pythagoras' theorem.

Read more about this topic: Pythagorean Theorem, Generalizations, Non-Euclidean Geometry

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