I found this question recently in my booklet on hyperbolic geometry asking a very simple question but I could not answer it:
Why can we not define the area of a hyperbolic triangle as in the plane as half the product of the perpendicular and the base?
I know the half plane model and the Poincare disk models but I cannot find a satisfactory explanation. I thought it might have something to do with that there are no rectangles in hyperbolic geometry but I cannot proceed. Help appreciated.
What is the area? Well, at least we want it to be (1) a non-negative function of a polygon that is (2) additive: $S(A\cup B)=S(A)+S(B)$ if $A\cap B$ has no interior points. It turns out, these two simple properties define $S$ almost uniquely — it's unique up to multiplication by a constant.
Now in Euclidean geometry one can prove that $S(\Delta)=ah_a$ by considering a triangle inside the rectangle — but because 'there are no hyperbolic rectangle' the same proof doesn't work in hyperbolic geometry.