The Poincare disk is $\{ z \in \mathbb{C} : |z|<1 \}$ and the hyperbolic distance from the origin is given by $\rho = 2 \tanh^{-1}(r)$ where $r$ is the Euclidean distance between the point and the origin.
In most references I have come across, this is just given as a definition. I would like a more intuitive understanding about where it comes from.
Presumably we should be able to see a metric induced on the Poincare disk as a result of the stereographic projection of the hyperboloid? Is this the correct idea?