Question

Find the radius of the incircle of a regular hexagon of side $6cm$

Answer 1

Let $AB$ be the side of the regular hexagon and let $O$ be the centre of the incircle.
Draw $OC\perp AB$. If $r$ is the radius of the circle, then $OC=r$. So,
$\angle AOB=\frac{{360}^{\circ }}{6}={60}^{\circ }$
$\therefore \angle AOC=\frac{{60}^{\circ }}{2}={30}^{\circ }$
$\therefore \tan {30}^{\circ }=\frac{AC}{r}$
i.e., $\frac{1}{\sqrt{3}}=\frac{3}{r}$
$\therefore r=3\times 1.732=\mathrm{5..196}cm$
Hence, radius of incircle is $5.196cm$