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

Answer 1

Answer: (B)

$r=\frac{a}{2}cot\frac{\pi }{n}$
‘a’ is side of polygon.
$R=\frac{a}{2}cosec\frac{\pi }{n}$
$\frac{r}{R}=\frac{cot\frac{\pi }{n}}{cosec\frac{\pi }{n}}=cos\frac{\pi }{n}$
$cos\frac{\pi }{n}\ne \frac{2}{3}$ for any $n\in N.$