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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?

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2 Answers 2

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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.

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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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