How to calculate the radius of a circle inscribed in a regular hexagon? If I know how long one side of a regular hexagon is, what's the formula to calculate the radius of a circle inscribed inside it?
Illustration:


Draw the six isosceles triangles.
Divide each of these triangles into two right angled triangles.
Then you have
$s = 2x = 2 (r \sin \theta)$
where $r$ is the radius of the circle, $\theta$ is the top angle in the right angled triangles and there are in total $12$ of these triangles so its easy to figure out $\theta$. $x$ is the short side in these right angled triangles and $s$ is of course the outer side in the isosceles triangles, i.e. the side length you say you know. 
Hence the formula for the radius is
$$r = \frac{s}{2 \sin \theta}$$
