Question

The ratio of area of a regular polygon of $n$ sides inscribed in a circle to that of the polygon of same number of sides circumscribing the same circle is $3:4$. Then the value of $n$ is

• A. $6$
• B. $4$
• C. $8$
• D. $1$

Answer 1

The correct option is A $6$

Let $a$ be the radius of the circle, then the ratio of the area of regular polygons of $n$ sides inscribed to circumscribing the same circle is given by
$\frac{S_1}{S_2}=\frac{\frac{1}{2}n{a}^2\sin \left(2\pi /n\right)}{n{a}^2\tan \left(\pi /n\right)}=\frac{3}{4}\Rightarrow {\cos }^2\left(\frac{\pi }{n}\right)=\frac{3}{4}$
Or,
$\cos \left(\frac{\pi }{n}\right)=\frac{\sqrt{3}}{2}\frac{\pi }{n}=\frac{\pi }{6}\Rightarrow n=6$