# The Area of Spherical Polygons

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 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 $E=5,E=6,...,E\approx20$.

-
Consider a spherical rhombus R. Why should not two polygons differ in area by the area of R, and still share the same configuration of sides (excepting those sides involved in R)? I do not see how you can expect the area to depend only on the number of sides, even given all sides are the same length. Gerhard "Ask Me About System Design" Paseman, 2012.05.22 –  Gerhard Paseman May 23 '12 at 4:04
The area of a simple $n$-gon on the unit sphere is $(n-2)\pi - \sum_i \gamma_i$, where $\gamma_i$ is the $i$th internal angle. Since you know $\sum_i \gamma_i$, you're done. (I don't understand why the discussion at Wikipedia was so messy; the area of a regular spherical 4-gon with internal angle $\gamma$ is $2\pi-4\gamma$.) –  JeffE May 23 '12 at 7:46