Regular hyperbolic polygon with prescribed area
= Regular hyperbolic polygon with prescribed area
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.