Hyperbolic area ($dA=dxdy/y^2$)

The metric area of a hyperbolic surface. In the Poincare half-plane model the density is $dxdy/y^2$. A finite-area torsion-free quotient of genus $g$ with $r$ cusps has area $2\pi (2g-2+r)$ by the Gauss-Bonnet theorem.