Using metrics is it possible to derive the circumference and area of a circle in hyperbolic space. I've found that the answer (without using metrics) are: C=2πsinh(r) and A=4πsinh2(r/2). But I'm unclear if this r represents the actual proper distance or the radius in hyperbolic space.
2026-03-31 07:02:53.1774940573
Hyperbolic space and metrics
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According to your formula, if $r$ is $1$, then $A$ is finite. But $r=1$ in Euclidean distance cover the whole unit disc. But the hyperbolic area of disc is infinite. For a simple idea of the proof see Here.