Question

The sum of the radii of inscribed and circumscribed circles of an $n$ -sided regular polygon with side equal to one unit is?

• A. $\frac{1}{2}\cot \frac{\pi }{2n}$
• B. $\cot \frac{\pi }{2n}$
• C. $\cot \frac{\pi }{n}$
• D. $\frac{1}{2}\tan \frac{\pi }{2n}$

Answer 1

The correct option is A $\frac{1}{2}\cot \frac{\pi }{2n}$

From the figure:
Side of polygon $\left(AB\right)=1$
$AO=\frac{1}{2}$
$\angle O=\frac{\pi }{2n}$
In right angled $△COA$ :
$\sin O=\frac{AC}{AO}$
$\Rightarrow \sin \frac{\pi }{n}=\frac{1}{2R}$ ..(1)
$\tan O=\frac{AC}{CO}$
$\Rightarrow \tan \frac{\pi }{n}=\frac{1}{2r}$ ...(2)
From (1) and (2)
$R+r=\frac{1}{2}(\frac{1}{\sin \frac{\pi }{n}}+\frac{1}{\tan \frac{\pi }{n}})$
$\Rightarrow R+r=\frac{1}{2}\left(\frac{1+\cos \frac{\pi }{n}}{\sin \frac{\pi }{n}}\right)=\frac{1}{2}\left(\frac{2{\cos }^2\frac{\pi }{2n}}{2\cos \frac{\pi }{2n}\sin \frac{\pi }{2n}}\right)$
$\Rightarrow R+r=\frac{1}{2}\cot \frac{\pi }{2n}$
Ans: A