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  1. In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, then is there a similar thing do happen in Spherical and hyperbolic spaces?

  2. In Euclidean plane, we know that the cosine law says that (suppose $\alpha$, $\beta$, $\gamma$ are the angles, and $a, b, c$ are the lengths opposite $a, b, c$ respectively.) $ \cos\gamma=\frac{a^2+b^2-c^2}{2ab}. $ Then is there a analogue in spherical and hyperbolic geometry? I have noted that the First and Second Cosine law are not so clearly relevant with this formula.

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    For hyperbolic space, the area of a triangle is $\pi - (\alpha + \beta + \gamma)$. So a triangle consisting of three hyperparallel lines asymptotic to each other is $\pi$, and all other triangles are smaller.2011-10-26

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Yes,there are analogues in spherical and hyperbolic geometry.

Start with a right angled triangle $\gamma=\pi/2$.

$\cos(c/R)=cos(a/R) cos(b/R) $

can be expanded in power series:

$( 1-(c/R)^2/2)\approx (1-(a/R)^2/2)*(1-(b/R)^2/2) $

and let R go to infinity to derive Pythagoras theorem approaching from spherical side getting

$ c^2 = a^2 + b^2 $

Likewise,

$\cosh(c/R)=\cosh(a/R) \cosh(b/R) $, expand cosh, let R go to infinity to derive it approaching from hyperbolic side:

$ c^2 = a^2 + b^2 $

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    Gro-Tsen's article http://www.madore.org/~david/weblog/d.2013-12-17.2175.trigonometrie-triangle.html#d.2013-12-17.2175 expands on these analogs in more detail.2018-12-31