Question

Find the length of the side of a regular polygon of $9$ sides inscribed in a circle of radius $8$ units

Answer 1

Let $AB$ be a side of the rectangular polygon with $9$ sides in the circle of radius $8$ units.
If $O$ is a centre of the circle, then $\angle AOB=\frac{{360}^{\circ }}{9}={40}^{\circ }$. Draw $OC\perp AB$ then
$\angle AOC=\frac{{40}^{\circ }}{2}={20}^{\circ }$
$\sin {20}^{\circ }=\frac{AC}{OA}=\frac{AC}{8}$
i.e., $0.3420=\frac{AC}{8}$
$AC=0.3420\times 8=2.736$
$\therefore$ Length of the side $AB=2\times AC=2\times 2.736=5.472units$