Help in a mathematical relation I'm trying to give my user the possibilty to get a value using a mathematical relation.
I need your help in this one.
What it is the value of $A$?
$$ S = \frac{{\color{red}A}B^2}{4\tan\frac{\pi}{{\color{red}A}}} $$

As was pointed out in a comment, the equation is of a type that typically does not have a closed form solution. That leaves numerical procedures. But there is more to be said, that might be useful. 
Let $x=\frac{\pi}{A}$. Then we can rewrite our equation as $\cot x=kx$, where $k$ is a constant easily obtained from our parameters $B$ and $S$. A sketch of $y=\cot x$ and $y=\cot x$ shows that there are infinitely many solutions, and that large solutions $x$ are near a multiple of $\pi$. (In terms of $A$, small solutions $A$ are close to the reciprocal of an integer.)
This brings us close to a very similar problem that has been much discussed, finding good estimates for large solutions of $\tan x=x$. We can get good general estimates for $\tan x$ when $x$ is close to an odd multiple of $\frac{\pi}{2}$. Essentially identical estimates can be made for $\cot x$ near multiples of $\pi$. So even though one cannot expect a closed form solution of the original equation, one can get good estimates of small solutions $A$ in terms of the parameters.
