Annulus between the incircle and circumcircle of a regular hexagon

Text

A regular hexagon has side $8$ cm. Compute the area of the annulus between the inscribed and circumscribed circles of the hexagon.

Solution

For a regular hexagon of side $\ell$: circumradius $R=\ell=8$, apothem (inradius) $r=\dfrac{\ell\sqrt3}{2}=4\sqrt3$. $\mathcal A=\pi(R^2-r^2)=\pi(64-48)=16\pi\approx50.27\ \text{cm}^2.$ $\boxed{\mathcal A=16\pi\approx50.27\ \text{cm}^2}$