Mathematics - Properties of Triangle Question with Solution | TestHub

For a regular $n$ -sided polygon, let $r$ be its inradius and $R$ be its circumradius.

The relationship between the inradius and circumradius of a regular $n$ -sided polygon is given by $r=R\cos \left(\frac{\pi }{n}\right)$. Given that the ratio of its inradius to its circumradius is $\frac{\sqrt{3}}{2}$, we have $\frac{r}{R}=\frac{\sqrt{3}}{2}$. Therefore, $\cos \left(\frac{\pi }{n}\right)=\frac{\sqrt{3}}{2}$. We know that $\cos \left(\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}$. Comparing the arguments, we get $\frac{\pi }{n}=\frac{\pi }{6}$, which implies $n=6$.