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



Formula for the Perimeter of a spherical triangle?
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. 


