Question

If the perimeter of a circle is equal to the perimeter of a regular polygon of 'n' sides, then their areas are in the ratio:

• A. $\tan \left(\frac{\pi }{n}\right):\frac{\pi }{n}$
• B. $\cos \left(\frac{\pi }{n}\right):\frac{\pi }{n}$
• C. $\sin \left(\frac{\pi }{n}\right):\frac{\pi }{n}$
• D. $\cot \left(\frac{\pi }{n}\right):\frac{\pi }{n}$

Answer 1

Answer: (A)

$\tan \left(\frac{\pi }{n}\right):\frac{\pi }{n}$

Perimeter of a circle is equal to the perimeter of a regular polygon

$2\pi r=n\times a$

where $r=$ radius of circle, $a=$ side of polygon with $n$ sides

$\frac{r}{n}=\frac{a}{2\pi }$

Area of Circle = $\pi r^2$

Area of Polygon = $\frac{n{a}^2}{4tan\left(\frac{\pi }{n}\right)}$

Ratio $=\frac{\pi r^2}{n{a}^2}\times 4\tan \left(\frac{\pi }{n}\right)$

$=\frac{\pi }{a^2}\times \frac{n{a}^2}{4{\pi }^2}\times 4\tan \frac{\pi }{n}$ ......... $[\frac{r^2}{n^2}=\frac{a^2}{4{\pi }^2}]$

$=\frac{\tan \pi }{n}:\frac{\pi }{n}$

Hence, $\tan \left(\frac{\pi }{n}\right):\frac{\pi }{n}$ is the correct answer.