Hyperbolic area
= Hyperbolic area
{title2=$dA=dx\,dy/y^2$}
The metric area of a <hyperbolic surface>. In the <Poincare half-plane model> the density is $dx\,dy/y^2$. A finite-area torsion-free quotient of <genus> $g$ with $r$ cusps has area $2\pi(2g-2+r)$ by the <Gauss-Bonnet theorem>.