Question

The area of the circle and the area of the regular polygon of n sides and of perimeter equal to that of circle are in the ratio of

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$\text{Let r be the radius of the circle}\therefore \text{Its are}{A}_1=\pi r^2\text{Length of one side of a regular polygon of n sides}=\text{perimeter of the circle}/n\frac{2\pi r}{n}=a\left(say\right)\therefore \text{Area of the polygon}=A_2=\frac{1}{4}\pi a^{2}cot\left(\frac{\pi }{n}\right)=\frac{{\pi }^2{r}^2}{n}cot\left(\frac{\pi }{n}\right)\therefore A_1;A_2=\pi r^2;\frac{{\pi }^2{r}^2}{n}cot\left(\frac{\pi }{n}\right)=tan\left(\frac{\pi }{n}\right):\frac{\pi }{n}$