Area of a hyperbolic disc

In curvature minus one, a hyperbolic disc of radius $R$ has areaIn the Poincare disc this follows by integrating $4r(1-r^2{)}^{-2}drd\theta$ up to Euclidean radius $\tanh (R/2)$.

Hyperbolic discs of area $\pi /2$ have radius $\log 2$. Starting at the origin and placing their centers successively along one radial geodesic gives center distance $n\log 4$ and Euclidean coordinate