Hyperbolic polygon area

For a geodesic hyperbolic $n$-gon of curvature $-1$ with interior angles ${\alpha }_1,\dots ,{\alpha }_n$, triangulation or the Gauss--Bonnet theorem gives

In curvature $-1$, a regular $n$-gon of circumradius $R>0$ has interior angle $\alpha (R)=2\arctan (\cot (\pi /n)/\cosh R)$, by the right-triangle angle form of the hyperbolic law of cosines. Thus its hyperbolic polygon area is $(n-2)\pi -n\alpha (R)$, continuously and strictly increasing from $0$ to $(n-2)\pi$. Every area in that open interval is therefore attained at exactly one circumradius.