TITLE: Formula for the Perimeter of a spherical triangle?
QUESTION [10 upvotes]: 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

REPLY [15 votes]: 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.