Question

For a regular polygon, let $r$ and $R$ be the radii of the inscribed and the circumscribed circles. A false statement among the following is -

• A. There is regular polygon with $\frac{r}{R}=\frac{1}{2}$
• B. There is regular polygon with $\frac{r}{R}=\frac{1}{\sqrt{2}}$
• C. There is regular polygon with $\frac{r}{R}=\frac{2}{3}$
• D. There is regular polygon with $\frac{r}{R}=\frac{\sqrt{3}}{2}$

Answer 1

The correct option is C There is regular polygon with $\frac{r}{R}=\frac{2}{3}$

From diagram I : $A=\frac{2\pi }{n}\Rightarrow \frac{a}{2R}=\frac{sin\pi }{n}$

From diagram II : $A=\frac{\pi }{n}\Rightarrow \frac{a}{2r}=\frac{tan\pi }{n}$

$\frac{r}{R}=\frac{cos\pi }{n}\Rightarrow n=3\Rightarrow \frac{r}{R}=\frac{1}{2}$

$n=4\Rightarrow \frac{r}{R}=\frac{1}{\sqrt{2}}$

$n=6\Rightarrow \frac{r}{R}=\frac{\sqrt{3}}{2}$