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 $n \in N.$