Question

The sum of the radii of inscribed and circumscribed circles of an n sided regular polygon of side ${}^{\prime }{a}^{\prime }$ is

• A. $=\frac{a}{2}(\frac{1}{\sin \pi /2x}+\cot \frac{\pi }{x})$
• B. $=\frac{a}{2}(\frac{1}{\sin \pi /x}+\cot \frac{\pi }{2x})$
• C. $=\frac{a}{2}(\frac{1}{\sin \pi /x}+\cot \frac{\pi }{x})$
• D. None of these

Answer 1

Answer: (C)

$=\frac{a}{2}(\frac{1}{\sin \pi /x}+\cot \frac{\pi }{x})$

$R\sin \theta =\frac{a}{2}$
$R=\frac{a}{2\sin \theta }$
$\tan \theta =\frac{a/2}{r}$
$r=\frac{a}{2\tan \theta }\theta =\frac{2\pi }{n}\times \frac{1}{2}$
$R+r=\frac{a}{2}(\frac{1}{\sin \theta }+\frac{\sin \theta }{\cos \theta })=\frac{\pi }{x}$
$=\frac{a}{2}(\frac{1}{\sin \pi /x}+\cot \frac{\pi }{x})$