I have read numerous paper over area calculation in hyperbolic geometry but just can't seem to understand how to calculate a triangle's area in hyperbolic geometry. It would be nice to have a proof too. Thanks in advance.
Edit: I did some more research since and I now understand how you can get the area using constant K and the defect of the triangle. However now I am confused about how you are suppose to obtain constant K. Do you NEED the area and the defect of at least one triangle to get the constant or is there any other way to calculate it?
Edit: Also, the formula on wikipedia contains a variable R, which I assume means radius. So how can one find R or is it usually 1 like K?
The problem is a bit that most models of the hyperbolic plane have the build in assumption that the curvature of the surface = -1.
I did ask previously questions about this same question but never got a good answer,
But now i think there are some ways:
For example construct a equilateral triangle with sides length $m$ and angles $\alpha$ (radians)
Then the absolute length of a side (so with a curvature of -1 ) is:
$$ arcosh (\frac {\cos^2\alpha +\cos\alpha }{ \sin^2 \alpha } ) $$
so K is then $m$ divided by the calculated length .
Ps 1 : This just a rough idea, maybe I made some mistake somewhere. and probably there are better ways. (or with smaller triangles, you need quite big triangles to get a measurable angle that is not $60^\circ $ )
Ps 2: I don't think area measures are a good starting point to calculate curvature. Area is in hyperbolic geometry a tricky (multiply the area of a triangle by 4 and the shape of the triangle changes) , therefore in the above I stuck to length and angle measures. they are much less tricky
Comments invited , Maybe we can find together a better way :)