# Area of a Regular Polygon 3

Area of a regular polygon of n sides when the radius r of the circumscribed circle is given

Let $OA = R$ be the radius of the circumscribed circle.

$\therefore$ the area of the polygon $= n \times$ area of $\Delta AOB$

But the area of $\Delta AOB = \frac{n}{2} \times OB \times OA \times \sin \frac{{{{360}^ \circ }}}{n}$

Also, the perimeter of the polygon $= n{\text{ }}AB$

But $\frac{{AD}}{{OA}} = \sin \frac{{{{180}^ \circ }}}{n}$
$\therefore$ $AD = OA\sin \frac{{{{180}^ \circ }}}{n}$

Also $AD = \frac{{AB}}{2}$
$\therefore$ $AB = 2AD$
$\therefore$ $AB = 2AD = 2OA\sin \frac{{{{180}^ \circ }}}{n}$
$\therefore$ Perimeter $= nAB$

Or $P = n \times 2OA\sin \frac{{{{180}^ \circ }}}{n} = 2nR\sin \frac{{{{180}^ \circ }}}{n}$ (As$OA = R$)

Example:

A regular decagon is inscribed in a circle, the radius of which is$10cm$. Find the area of the decagon.

Solution:

Here $n = 10$, $R10cm$

The area of the decagon $= \frac{{n{R^2}}}{2}\sin \frac{{{{360}^ \circ }}}{n} = \frac{{10{{(10)}^2}}}{2}\sin \frac{{{{360}^ \circ }}}{2}$

The area of the decagon $= 500\sin {36^ \circ } = 500(0.5878) = 293.9$ square cm.