Question

The area of a circle is $A_1$, and the area of a regular pentagon inscribed in a circle is $A_2$ . Then $A_1:A_2$ is

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is B

In a $△OAB,OA=OB=r$
And $\angle AOB=\frac{{360}^o}{5}={72}^o$
$\therefore ar\left(△OAB\right)=\frac{1}{2}.r.r.sin{72}^o=\frac{1}{2}.r^{2}cos{18}^o$
$\Rightarrow ar\left(pentagon\right)=\frac{5}{2}.r^{2}cos{18}^o$
$\therefore A_1:A_2=\frac{2\pi r^2}{5r^{2}cos{18}^o}=\frac{2\pi }{5}sec\frac{\pi }{10}$