Question

If a regular pentagon and regular decagon have the same perimeter and $A_1,A_2$ are areas of regular pentagon and regular decagon respectively then $\frac{A_1}{A_2}$ is

• A. $\frac{4}{\sqrt{5}}$
• B. $\frac{\sqrt{5}}{4}$
• C. $\frac{2}{\sqrt{5}}$
• D. $\frac{\sqrt{5}}{2}$

Answer 1

The correct option is C $\frac{2}{\sqrt{5}}$

Area of polygon $=n\frac{a^2}{4}\cot \frac{\pi }{n}$ where $a$ is side of regular polygen
$\left[\begin{array}{c}[5a_1=10a_2(\text{given}\left)\right]\\ a_1=\text{length of penttagon}\\ a_2=\text{length of decagon}\end{array}\right]$
$A_1=5\frac{a_1^2}{4}\cot \frac{pi}{5}$
$A_2=10\frac{a_2^2}{4}\cot \frac{\pi }{10}$
$\frac{A_1}{A_2}=\frac{2}{\sqrt{5}}$