Question

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

Answer 1

Hint: Firstly we use the expression to find the area of regular polygon with ${\text{n}}$ sides inscribed i.e. $\dfrac{1}{2}n{a^2}\sin \left( {\dfrac{{2\pi }}{n}} \right)$ and circumscribing the curve, which is $n{a^2}\tan \left( {\dfrac{\pi }{n}} \right)$, so, we can take their ratio and equate to the given ratio. After simplification, we can solve for ${\text{n}}$ to get the required answer.

Complete step by step answer:

Area of regular polygon of ${\text{n}}$ sides inscribed in a circle is given by,
${A_1} = \dfrac{1}{2}n{a^2}\sin \left( {\dfrac{{2\pi }}{n}} \right)$
The area of regular polygon of ${\text{n}}$sides circumscribing a circle is given by,
${A_2} = n{a^2}\tan \left( {\dfrac{\pi }{n}} \right)$
We can take the ratio of the two areas,
$\dfrac{{{A_1}}}{{{A_2}}} = \dfrac{{\dfrac{1}{2}n{a^2}\sin \left( {\dfrac{{2\pi }}{n}} \right)}}{{n{a^2}\tan \left( {\dfrac{\pi }{n}} \right)}}$
Using the identity $\sin 2A = 2\sin A\cos A$and \[\tan A = \dfrac{{\sin A}}{{\cos A}}\], we get
\[\dfrac{{{A_1}}}{{{A_2}}} = \dfrac{{\dfrac{1}{2}n{a^2}2\sin \left( {\dfrac{\pi }{n}} \right)\cos \left( {\dfrac{\pi }{n}} \right)}}{{n{a^2}\dfrac{{\sin \left( {\dfrac{\pi }{n}} \right)}}{{\cos \left( {\dfrac{\pi }{n}} \right)}}}}\]
Canceling the common terms, we get,
\[\dfrac{{{A_1}}}{{{A_2}}} = {\cos ^2}\left( {\dfrac{\pi }{n}} \right)\]
It is given that the ratio of this area is ${\text{3:4}}$. So,
\[\dfrac{3}{4} = {\cos ^2}\left( {\dfrac{\pi }{n}} \right)\]
\[
   \Rightarrow \cos \left( {\dfrac{\pi }{n}} \right) = \dfrac{{\sqrt 3 }}{2} \\
   \Rightarrow \dfrac{\pi }{n} = {\cos ^{ - 1}}\dfrac{{\sqrt 3 }}{2} = \dfrac{\pi }{6} \\
 \]
By solving we get,
$n = 6$
So, the value of n is 6.
Therefore, the correct answer is option A.

Note: The equations used here are the equations to find the area of regular polygon with ${\text{n}}$ sides inscribed i.e. $\dfrac{1}{2}n{a^2}\sin \left( {\dfrac{{2\pi }}{n}} \right)$ and circumscribing the curve, which is $n{a^2}\tan \left( {\dfrac{\pi }{n}} \right)$, radius a. We also use trigonometric identities in the problem for the solving part. The basic trigonometric values must be known. We must take care of the order while taking the ratio.