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:

 A: Label the center of the circle.  Draw six lines from the the center to the circle to the vertices of the hexagon.  (These lines will be longer than the radius.)  This will divide the circle into six triangles.
Question for you: Tell me every thing you can about these triangles.  In particular, what are the lengths of the lines from the center?
Now draw six radii of the circle to the six edges of the hexagon.  Along with the six "spokes" before you have divided the hexagon into twelve triangles.
Question for you: tell me every thing you can about these triangles.  In particular:
are they congruent to each other?
what are the angles of these triangles?
What are the lengths of the sides of these triangles?
And from there I will ask you these two questions:  What is the radius of the circle?  and, what is the formula for the area of the circle.
A: The radius equals the height of the equilateral triangles of side $s$.
By Pythagoras,
$$h^2+\left(\frac s2\right)^2=s^2$$ so that
$$h=\frac{\sqrt 3}2s.$$
A: 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
$$r = \frac{s}{2 \sin \theta}$$
A: A regular Hexagon can be split into $6$ equilateral triangles. Since the inscribed circle is tangent to the side lengths of the Hexagon, we can draw a height from the center of the circle to the side length of the Hexagon.
Using the $30-60-90$ rule, the height is $\frac {x\sqrt{3}}{2}$ with a Hexagon with a side length of $x$ units.
So the radius of the circle is $\frac {x\sqrt{3}}{2}$ with $x$ as a side length of the Hexagon.
*NOTE: This is only true when the Hexagon is a regular Hexagon!
And for the area of the circle, just use the formula for the area of a circle ($A=\pi r^2$) where $r$ is the radius.
A: Using the Pythagorean theorem you can turn the hexagon into 12 30-60-90 triangles or 6 equilateral triangles
Hypotenuse squared - half the hypotenuse squared = the length of the radius of the circle (squared) 
Diagram. The red lines are the hypotenuses and the yellow lines are radii of the circle.
