Question

The ratio of the area of a regular polygon of $n$ sides inscribed in a circle to that of the polygon of same number of sides circumscribing the same is 3:4. Then the value of $n$ is?
A. 6
B. 4
C. 8
D. 12

Answer 1

Hint:
We will write the areas of the polygon when inscribed in a circle and when the polygon is circumscribed in a circle. Then, we will write it equal to the given ratio. Simplify the expression using trigonometric identities and solve the equation to find the value of $n$.

Complete step by step solution:
As we know that the area of the polygon of $n$ sides inscribed in a circle is given as ${A_1} = \dfrac{1}{2}n{a^2}\sin \left( {\dfrac{{2\pi }}{n}} \right)$
Also, the area of the regular polygon of $n$ sides circumscribing a circle is given as ${A_2} = n{a^2}\tan \left( {\dfrac{\pi }{n}} \right)$
Then, we will take the ratio of the two ratios.
$\dfrac{{{A_1}}}{{{A_2}}} = \dfrac{{\dfrac{1}{2}n{a^2}\sin \left( {\dfrac{{2\pi }}{n}} \right)}}{{n{a^2}\tan \left( {\dfrac{\pi }{n}} \right)}}$
Use the identity $\sin 2x = 2\sin x\cos x$ and $\tan x = \dfrac{{\sin x}}{{\cos x}}$ to simplify the given terms.
$\dfrac{{{A_1}}}{{{A_2}}} = \dfrac{{\dfrac{1}{2}n{a^2}2\sin \left( {\dfrac{\pi }{n}} \right)\cos \left( {\dfrac{\pi }{n}} \right)}}{{n{a^2}\dfrac{{\sin \left( {\dfrac{\pi }{n}} \right)}}{{\cos \left( {\dfrac{\pi }{n}} \right)}}}}$
After cancelling the terms, we will get,
$\dfrac{{{A_1}}}{{{A_2}}} = {\cos ^2}\left( {\dfrac{\pi }{n}} \right)$
We are given that the ratio the above areas is 3:4
This implies,
$
  \dfrac{3}{4} = {\cos ^2}\left( {\dfrac{\pi }{n}} \right) \\
   \Rightarrow \cos \left( {\dfrac{\pi }{n}} \right) = \dfrac{{\sqrt 3 }}{2} \\
   \Rightarrow \dfrac{\pi }{n} = {\cos ^{ - 1}}\dfrac{{\sqrt 3 }}{2} \\
   \Rightarrow \dfrac{\pi }{n} = \dfrac{\pi }{6} \\
$
By solving the above expression, we will get,
$n = 6$

Hence, option A is correct.

Note:
One must know that the formula of area of regular polygon with sides $n$ inscribed in a circle is $\dfrac{1}{2}n{a^2}\sin \left( {\dfrac{{2\pi }}{n}} \right)$ and when the polygon is circumscribed in a circle is $n{a^2}\tan \left( {\dfrac{\pi }{n}} \right)$. Students must have the basic trigonometric values for these questions. Also, one must take care of the order of the ratio.