I ve been looking in wikipedia and other sites for "hyperbolic angle", but it is not drawn anywhere. Only an area is shaded everywhere. Is it even possible to draw it?
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An angle is formed between two rays that originate at a common point. This is true in both Euclidean and hyperbolic geometries, so the question is more, from a given drawing (or two given rays), how would you measure the angle?
In Euclidean geometry, you can derive an area formula that is $A = r^2 \theta/2$, and so the area is directly proportional to the angle. Simply draw an arbitrary circular arc centered on the starting point of the rays, and there you go.
I must admit not knowing or being able to find the area of a hyperbolic sector as a function of angle, but I suspect (from a similar argument to circular sectors), it would also be $A=r^2 \theta/2$, just that $r$ is constant on a hyperbola.
So the reason areas are shaded is because the areas are directly proportional to the actual angles, and this is the only meaningful way to distinguish between a Euclidean and hyperbolic angle between two drawn rays.
The angle at $(0,0)$ of the shaded area is the hyperbolic angle: