Question

If a regular pentagon and regular decagon have the same perimeter, then their areas are as $2$ to $\sqrt{5}$

• A. True
• B. False

Answer 1

The correct option is A True

Let the perimeter of the pentagon and decagon be $10c,$
then each side of the pentagon is $2c$
and its area is $\frac{5\times {\left(2c\right)}^2}{4}\cos \left(\frac{\pi }{5}\right)$ $\because$ Area of polygon$=\frac{n{a}^2}{4}\cot \left(\frac{\pi }{n}\right)$
Each side of the decagon is $c$
and its area is $\frac{10\times c^2}{4}\cot \left(\frac{\pi }{10}\right)$
$\therefore$Area of Pentagon$:$Area of Decagon
$=2\cot {36}^0:\cot {18}^0$
$=\frac{2\cot {36}^0}{\cot {18}^0}$
$=\frac{2\cos {36}^0\sin {18}^0}{\sin {36}^0\cos {18}^0}$
$=\frac{2\cos {36}^0}{2{\cos }^2{18}^0}$
$=\frac{2\cos {36}^0}{1+\cos {36}^0}$
$=\frac{\frac{2(\sqrt{5}+1)}{4}}{(1+\frac{\sqrt{5}+1}{4})}$
$=\frac{2(\sqrt{5}+1)}{4}\times \frac{4}{5+\sqrt{5}}$
$=\frac{2}{\sqrt{5}}$
$=2:\sqrt{5}$