Let . A metric space is hyperbolic if, for any figure in that is a geodesic triangle with respect to and for every , there exists a point such that .

A hyperbolic metric space is a metric space that is hyperbolic for some .

Although a metric space is hyperbolic if it is hyperbolic for some , one usually tries to find the smallest value of for which a hyperbolic metric space is hyperbolic.

A simple example of a hyperbolic metric space is the real line under the usual metric. Given any three points , we always have that . Thus, for any , we can take . Therefore, the real line is 0 hyperbolic. Similar reasoning can be used to show that every real tree is 0 hyperbolic.