Question

If $A_1$ and $A_2$ be the areas of two regular polygons having the same perimeter and number of sides be $n$ and $2n$ respectively, then $\frac{A_1}{A_2}$ is -

• A. independent of $n$
• B. depends on $n$
• C. is a function of $\cos \left(\frac{\pi }{n}\right)$ only with no fractional powers
• D. is a function of $\sin \left(\frac{\pi }{n}\right)$ only with no fractional powers.

Answer 1

Answer: (B), (C)

The correct options are
B depends on $n$
C is a function of $\cos \left(\frac{\pi }{n}\right)$ only with no fractional powers

Area of polygon with $n$ sides of length $x$ is,
$A=\frac{1}{4}.n{x}^2\cot \frac{\pi }{n}$
So, $A_1=\frac{1}{4}n{a}_1^2\cot \frac{\pi }{n},A_2=\frac{1}{4}\left(2n\right)a_2^2\cot \frac{\pi }{2n}$
& $n{a}_1=2n{a}_2$
$\frac{A_1}{A_2}=\frac{1}{2}\times 4\frac{\cot \left(\pi /n\right)}{\cot \left(\pi /2n\right)}=\frac{2\cos \frac{\pi }{n}.\sin \frac{\pi }{2n}}{\sin \frac{\pi }{n}.\cos \frac{\pi }{2n}}$
$=\frac{2\cos \frac{\pi }{n}.\sin \frac{\pi }{2n}}{2\sin \frac{\pi }{2n}\cos \frac{\pi }{2n}.\cos \frac{\pi }{2n}}$
$=\frac{2\cos \frac{\pi }{n}}{(1+\cos \frac{\pi }{n})}$