Question

If $r$ and $R$ are respectively the radii of the inscribed and circumscribed circle of a regular polygon of n sides such that $\frac{R}{r}=\sqrt{5}-1$, then $n$ is equal to?

• A. $5$
• B. $6$
• C. $10$
• D. none of these

Answer 1

The correct option is A $5$

from the fig:
given that: $\frac{R}{r}=\sqrt{5}-1$
In right angled $\Delta ACO$
$\cos \frac{\pi }{n}=\frac{1}{\sqrt{5}-1}=\frac{\sqrt{5}+1}{4}$
$\Rightarrow \cos \frac{\pi }{n}=\cos \frac{\pi }{5}$
Therefore, $n=5$
Ans: A