A regular hexagon is inscribed in a circle of radius $r$. Find the length of the sides of the hexagon and the distance of each side from the centre of the circle.

In a regular hexagon inscribed in a circle of radius $r$, the vertices divide the circle into 6 equal arcs, each subtending $\frac{360^\circ}{6} = 60^\circ$ at the centre.

For each side, the triangle formed by the centre and the two endpoints of the side is isosceles (two sides equal to $r$) with vertex angle $60^\circ$.
Hence the triangle is equilateral, and the side $= r$.
Length of each side of the hexagon $= r$.
Using Baudhāyana–Pythagoras theorem:

$$\begin{aligned} d^2 &= r^2 - \left(\frac{r}{2}\right)^2 \\ &= r^2 - \frac{r^2}{4} = \frac{3r^2}{4} \\ d &= \left(\frac{\sqrt{3}}{2}\right)r \end{aligned}$$

So, distance of each side from the centre is $\left(\frac{\sqrt{3}}{2}\right)r$.