Question

If the circle and the regular polygon of n sides are of perimeter equal then areas of circle and regular polygon of n sides are in the ratio of

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

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