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?
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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.
