Question

If $r$ and $R$ are respectively the radii of the inscribed and circumscribed circles 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. $18$

Answer 1

The correct option is A $5$

Let $a$ be the length of side of regular polygon then
$R=\frac{a}{2}\csc \left(\frac{\pi }{n}\right)$ and $r=\frac{a}{2}\cot \left(\frac{\pi }{n}\right)$
$\therefore \frac{R}{r}=\frac{\csc \left(\frac{\pi }{n}\right)}{\cot \left(\frac{\pi }{n}\right)}=\frac{\frac{1}{\sin \left(\frac{\pi }{n}\right)}}{\frac{\cos \frac{\pi }{n}}{\sin \frac{\pi }{n}}}=\frac{1}{\cos \left(\frac{\pi }{n}\right)}$
$\therefore \cos \left(\frac{\pi }{n}\right)=\frac{1}{\sqrt{5}-1}$
On rationalising the denominator, we get
$\cos \left(\frac{\pi }{n}\right)=\frac{1}{\sqrt{5}-1}=\frac{1}{\sqrt{5}-1}\times \frac{\sqrt{5}-1}{\sqrt{5}+1}=\frac{\sqrt{5}+1}{4}=\cos \left(\frac{\pi }{5}\right)$
On comparing $\cos \left(\frac{\pi }{n}\right)=\cos \left(\frac{\pi }{5}\right)$ we get $n=5$