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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.

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    $\begingroup$ I began to studied hyperbolic geometry three weeks ago, so I am not sure for the exact reason. But as far as I known, if we defined the area in hyperbolic plane as we did in Euclid plane, then at least the möbius transformations would not preserve the area as we except. I am really new at this topic, so I am not sure for my commenting. $\endgroup$ – KWSK Oct 2 '15 at 7:12
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    $\begingroup$ Nice question, but above my knowledge, i made a relatd question of a (tiny) bit simpler problem see math.stackexchange.com/questions/1462778/… $\endgroup$ – Willemien Oct 3 '15 at 18:07
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    $\begingroup$ Well, one reason is that in hyperbolic geometry $ah_a\neq bh_b$, so this 'area' would depend on the side we choose as the base — so in a sense it's not well-defined. $\endgroup$ – Grigory M Oct 3 '15 at 19:25
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    $\begingroup$ @KWSK Well, (hyperbolic) lengths of both the base and the (hyperbolic) perpedicular are manifestly invariant under hyperbolic motions. $\endgroup$ – Grigory M Oct 3 '15 at 19:27
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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.

triangle inside rectangle

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  • $\begingroup$ Thank you for a very clear well written answer $\endgroup$ – kroner Oct 3 '15 at 19:37
  • $\begingroup$ When scale factor for length is non-linear and varies point to point in either of these models what meaning can be there for an area, using a euclidean yardstick ? $\endgroup$ – Narasimham Oct 3 '15 at 19:50

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