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In the curvature-minus-one Poincare half-plane model, a geodesic polygon with $n$ sides and interior angles ${\alpha }_1,\dots ,{\alpha }_n$ has areaThis is the polygonal Gauss-Bonnet theorem.

A constant dilaton contributes ${\Phi }_0\chi (\Sigma )$ to the Euclidean worldsheet action by the Gauss-Bonnet theorem. With string coupling $g_s=e^{{\Phi }_0}$ and Euler characteristic $\chi =2-2g$, the path-integral weight is $g_s^{2g-2}$. Canonically normalized external closed-string vertices add one power of $g_s$ each.

With Euclidean curvature coupling $(4\pi {)}^{-1}\int \sqrt{h}\phi R$, a constant dilaton contributes ${\phi }_0\chi$ by the Gauss-Bonnet theorem. The genus-$g$ closed-string weight is $e^{-{\phi }_0\chi }=g_s^{2g-2}$ with $g_s=e^{{\phi }_0}$. If the coefficient is instead written as $\int \sqrt{h}\Phi R$, then in the same convention $\phi =4\pi \Phi$. The field normalization must be fixed before reading off the coupling.

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.