Question

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

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

Answer 1

The correct option is A

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

Explanation for correct answer:

Step 1: Finding $A_1$ and $A_2$

It is given that, the perimeter of a circle is equal to the perimeter of a polygon.

Let '$r$‘ is the radius of a circle and ’$a$' be the side of a regular polygon.

Therefore,

$2\mathrm{\pi r}=\mathrm{na}$ (’$n$' is the number of sides of a polygon)

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

Let the area of a circle is denoted by ‘$A_1$’

Therefore, $A_1={\mathrm{\pi r}}^2\to \left(i\right)$

Let '$A_2$' represent the area of a regular polygon, and the area of a polygon is calculated by using the formula given below.

$A_2=\frac{1}{4}n\times a^2\times \cot \left(\frac{\mathrm{\pi }}{n}\right)\to \left(ii\right)$

Substituting the value of $a=\frac{2\mathrm{\pi r}}{n}$ in equation $\left(ii\right)$ we get,

$$\begin{gathered} A_2=\frac{1}{4}n\times {\left(\frac{2\mathrm{\pi r}}{n}\right)}^2\times \cot \left(\frac{\mathrm{\pi }}{n}\right) \\ =\left({\frac{{\pi }^{2}r}{n}}^2\right)\times \cot \left(\frac{\mathrm{\pi }}{n}\right)\to \left(iii\right) \end{gathered}$$

Step 2: Determining the ratio of the two areas

By Dividing equation $\left(i\right)$ by equation $\left(iii\right)$ we get,

$$\begin{gathered} \frac{A_1}{A_2}=\frac{{\mathrm{\pi r}}^2}{\left({\frac{{\pi }^{2}r}{n}}^2\right)\times \cot \left(\frac{\mathrm{\pi }}{n}\right)} \\ =\frac{n}{\mathrm{\pi }\times \cot \left(\frac{\mathrm{\pi }}{n}\right)} \\ =\frac{n}{\mathrm{\pi }}\times \tan \left(\frac{\mathrm{\pi }}{n}\right) \end{gathered}$$

$\therefore A_1:A_2=\tan \left(\frac{\mathrm{\pi }}{n}\right):\frac{\mathrm{\pi }}{n}$

Hence, option (A) is the correct answer.