Paper 1, Section I, G

Geometry
Part IB, 2017

Give the definition for the area of a hyperbolic triangle with interior angles α,β,γ\alpha, \beta, \gamma.

Let n3n \geqslant 3. Show that the area of a convex hyperbolic nn-gon with interior angles α1,,αn\alpha_{1}, \ldots, \alpha_{n} is (n2)παi(n-2) \pi-\sum \alpha_{i}.

Show that for every n3n \geqslant 3 and for every AA with 0<A<(n2)π0<A<(n-2) \pi there exists a regular hyperbolic nn-gon with area AA.