circumscribe a regular polygon around a circle in hyperbolic geometry

Question

In the hyperbolic plane, let a circle of radius r be given. If we want to circumscribe a regular polygon with n sides around this circle (i.e., if we want the sides of the polygon to be tangents of the circle), how large must n be?

Answer 1

You can have a curvilinear equilateral triangle, square, pentagon &c $\dots$ just as in Euclidean geometry when the number of polygon side goes to infinity the area of circumscribed polygon approaches $ \pi r^2.$

Please try working on this as a starting point. At least for surfaces of constant Gauss curvature and constant geodesic curvature I believe that

if $n$ is the number of sides of a regular polygon on a surface of constant negative Gauss curvature $ -K $ and constant geodesic curvature $1/r$

$$ \sqrt{ -K} = k $$ then,

$$ \sinh ( k n r )/( k n) \rightarrow r, n \rightarrow \infty.$$