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Consider the ordinary sphere $\mathbb{S}^2\subset \mathbb{R}^3$ and a spherical triangle $T\subset \mathbb{S}^2.$ I'm looking for a formula from which the perimeter $P$ of $T$ is "computable" given the three interior angles $\gamma_1,\gamma_2,\gamma_3$ (and the area $\vert T \vert$) of the triangles, i.e. i'm searching for a formula

$$P=f(\gamma_1,\gamma_2, \gamma_3),$$

where $f$ is an explicitly given function (which is hopefully not too complicated).

Are there such formulas? I couldn't find any utilizable connections.

For instance there is a nice formula for the planar case given by the equation:

$\frac{P^2}{4\vert T \vert}= \sum_{i=1}^3 \cot \left( \frac{\gamma_i}{2}\right)$, where $P$ denotes the perimeter of the planar triangle.

I hope some experts can help me.

Best regards

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First, you don't need to know the area separately, since that is given by the classic formula $$ |T| = (\gamma_1+\gamma_2+\gamma_3) - \pi. $$ Second, if $\ell_i$ is the length of the side opposite $\gamma_i$, then the standard spherical trig formula called the polar law of cosines gives $$ \ell_i = \cos^{-1}\left(\frac{\cos\gamma_i + \cos\gamma_j\cos\gamma_k}{ \sin\gamma_j\sin\gamma_k}\right) $$ where $(i,j,k)$ is a permutation of $(1,2,3)$. Now, $$P = \ell_1+\ell_2+\ell_3$$ is such a formula.

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  • $\begingroup$ Thanks for your help! This is a nice formula. In my understanding this formula doesn't work for triangles with any of its lengths $\ell_i>\pi$. Do you agree or am I missing something? $\endgroup$
    – supersnail
    Commented Dec 1, 2014 at 15:08
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    $\begingroup$ @supersnail: I haven't checked such triangles, so I'm not sure. Clearly, the polar law of cosines doesn't work for the 'degenerate' triangles in which all three vertices lie on a single great circle. It might well be that you need to assume that all of the sides have length at most $\pi$. Have a look at the assumptions used to derive these formulae at en.wikipedia.org/wiki/Solution_of_triangles#Three_angles_given and also at en.wikipedia.org/wiki/Spherical_law_of_cosines . Alternatively, take a look at en.wikipedia.org/wiki/Half-side_formula, which might be better. $\endgroup$
    – Robert Bryant
    Commented Dec 1, 2014 at 16:03

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