Let a, b and c denote the cosh of the lengths of the sides of an hyperbolic triangle and A, B, and C its angles.

Its area is well knwon to be S = pi - A - B - C .

What is S in terms of a, b, c ?

In J. Smorodinskij, Fortschritte der Physik, 18 (1965) 157 -- 173

I find (without reference or proof)

cos(S/2) = (1 + a + b + c) / (4 (a' b' c')^2)

where a' is the cosh of half the lenght of the side a.

My cumbersum calculations yield

cos(S/2) = (1 + a + b + c) / (4 (a' b' c'))

Where do I find a simple proof of this simple formula?


2 Answers 2


One proof is sketched here: http://www.maths.gla.ac.uk/wws/cabripages/hyperbolic/harea2.html A more brute force way of expressing area through side lengths is to use the hyperbolic law of cosines.


Your version of the formula is correct. The proof can be found, for example, on pp. 102-103 in http://arxiv.org/abs/1102.0462 (The Hyperbolic Theory of Special Relativity, by J.F. Barrett).


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