Form a Circle with Circles I need to form a perfect circle out of circles. 
Given N number of circles each with radius R, how can I find the distance away from the center?
 A: Sorry, this is too brief, there should be a picture. Let the little circles all have radius $r$. Suppose there are $n$ of them, where $n\ge 3$.  Let $R$ be the distance from the centre of the big circle to the centre of each little circle. It turns out that 
$$R\sin\left(\frac{180^\circ}{n}\right)=r,\tag{$1$}$$
so now we can compute $R$.
To see that Formula $(1)$ is correct, draw two consecutive little circles, with centres $A$ and $B$ respectively. Let the big circle have centre $O$. By the definition of $R$, the big circle passes through $A$ and $B$. 
Drop a perpendicular from $O$ to the midpoint $M$ of $AB$. The two little circles touch at $M$. 
Note that $\angle AOM$ is $\dfrac{180^\circ}{n}$ and $AM=r$. The formula now follows by trigonometry.  
The question specifically asked not for $R$, but for the (nearest) distance from the centre of the big circle to the little circles. This is $R-r$.
Remark: I should have called the radius of the little circles $R$, to use the notation of the OP. But it is a little circle, so it should be $r$. Then one can reserve $R$ for the big one. 
