I am interested in finding a canonical general expression for the area of a spherical polygon in $\mathbb{S}^2$ knowing the side lengths of the polygon and a bound on the internal angles (we can assume a radius of $R=1$).


For what I am researching (I will not go into the background) I need the following conditions to be satisfied:


* The edge length of all spherical polygons in $\mathbb{S}^2$ is $\pi / 3$.
* The spherical polygons I want to consider may or may not be convex, in fact it is necessary that I be able to compute the area of a *non*-convex polygon in $\mathbb{S}^2$.


I know there are multiple ways to compute the area of a spherical triangle using the spherical law of cosines, L'Huilier's theorem, or other formulas, but I want to be more general than this. The type of information I know about the internal angles of the spherical polygons is a bound in terms of the degree of the vertex considered in a spherical simplicial $2$-complex $\mathcal{K}$ in $\mathbb{S}^2$. That is, labeling $\gamma\_{i}$ as an internal angle of the spherical polygon that
$$ \sum\limits\_{1 \leq i \leq b} \gamma\_{i} = \sum\limits\_{1 \leq i \leq b} (i-1)\arccos(\frac{1}{3})b\_{i}$$
where $b\_{i}$ denotes the number of vertices of degree $i$ in $\mathcal{K}$. The last comment about the internal angles may or may not be confusing, but I just wanted to mention that I know *something* about the internal angles of the spherical polygons. For an example of how difficult this problem may be, there was a [large discussion here](http://en.wikipedia.org/wiki/User%3aNorwegianBlue/area_of_a_square_on_the_surface_of_a_sphere) about determining the area of a spherical $4$-gon with given side length (and the answer was quite messy), so I'm hoping that some of you have ideas!


To summarize exactly what my question is, and what information I know:



> 
> You are given a number $E$ which tells you how many edges a spherical polygon $C$ in $\mathbb{S}^2$ has (all edges have length $\pi /3$ and $C$ is not necessarily convex). Determine the area of $C$ (or a function for the area of $C$).
> 
> 
> 


That is, I want to find the analogue in spherical geometry to the equations in Euclidean geometry which tell you the area of a regular polygon of a given number of sides. If such a general expression does not exist, I would be interested in the case for $E=5,E=6,...,E\approx20$.



The Area of Spherical Polygons
This excellent paper collects many useful formulae for spherical calculations, including (but hardly limited to) polygon area. Explanations are clear and well developed.


Some Algorithms for Polygons on a Sphere 
Robert.G.Chamberlain 
William.H.Duquette 
Jet Propulsion Laboratory


<http://hdl.handle.net/2014/40409>


* smh


