Question

The sum of the radii of inscribed and circumscribed circles for a $n$ sided regular polygon of side $a$ is

• A. $\frac{a}{2}\tan \frac{\pi }{2n}$
• B. $\frac{a}{2}\cot \frac{\pi }{4n}$
• C. $\frac{a}{2}\cot \frac{\pi }{2n}$
• D. $2a\cot \frac{\pi }{2n}$

Answer 1

The correct option is C $\frac{a}{2}\cot \frac{\pi }{2n}$

For a n sided regular polygon, If we join all the corners to the center of polygon, we get 'n' congruent isosceles triangles. Vertical angle of each such triangle is $\frac{2\pi }{n}$. The equal sides of the isosceles triangle are equal to the circumradius 'R' and the height is equal to inradius 'r'. (as shown in figure).
From the triangle, we get
$\tan \frac{\pi }{n}=\frac{a}{2r}$....(1)
$\sin \frac{\pi }{n}=\frac{a}{2R}$....(2)
From (1) and (2)
$R+r=\frac{a}{2}(\csc \frac{\pi }{n}+\cot \frac{\pi }{n})$
$=\frac{a}{2}\left(\frac{1+\cos \frac{\pi }{n}}{\sin \frac{\pi }{n}}\right)$
$=\frac{a}{2}\cot \frac{\pi }{2n}$