Question

A regular pentagon and a regular decagon have the same perimeter. Then the ratio of their areas are:

• A. $2:\sqrt{5}$
• B. $2:\sqrt{3}$
• C. $3:\sqrt{5}$
• D. $2:5$

Answer 1

The correct option is A $2:\sqrt{5}$

Let $AB$ be one of the $n$ sides of a regular polygon and $O$ be the centre of the circum circle.
If $AB=a$ then area of the polygon $=\frac{n}{2}AB.OD=\frac{n{a}^2}{4}\cot \frac{\pi }{n}$
Let the perimeter of pentagon and decagon be $10x$.
Then each side of the pentagon is $2x$ and its area is $5x^2\cot \frac{\pi }{5}$
(using equation $\left(i\right)$ where $n=5$ and $a=2x$)
Similarly, in a decagon, $n=10$ and $a=x$. Hence its area is $\frac{5}{2}{x}^2\cot \frac{\pi }{10}$
Thus, we have $\frac{\text{Area of pentagon}}{\text{Area of decagon}}=\frac{2\cot {36}^{\circ }}{\cot {18}^{\circ }}=\frac{2\cos {36}^{\circ }\sin {18}^{\circ }}{\sin {36}^{\circ }\cos {18}^{\circ }}=\frac{2\cos {36}^{\circ }\sin {18}^{\circ }}{2\sin {18}^{\circ }{\cos }^2{18}^{\circ }}=\frac{2\cos {36}^{\circ }}{2{\cos }^2{18}^{\circ }}=\frac{2\cos {36}^{\circ }}{1+\cos {36}^{\circ }}=\frac{2(\sqrt{5}+1)}{4(1+\frac{\sqrt{5}+1}{4})}=\frac{2(\sqrt{5}+1)}{5+\sqrt{5}}=\frac{2}{\sqrt{5}}$