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 6 (b) 4 (c) 8 (d) 12

Options

A. 6
B. 4
C. 8
D. 12

Explanation

To solve the problem, we need to find the value of \( n \) for which the ratio of the area of a regular polygon with \( n \) sides inscribed in a circle to that of the polygon circumscribing the same circle is \( \frac{3}{4} \).

### Step-by-Step Solution:

1. **Understanding the Areas**:
- Let \( A \) be the radius of the circle.
- The area \( S_1 \) of the regular polygon inscribed in the circle is given by:
\[ S_1 = \frac{1}{2} n A^2 \sin\left(\frac{2\pi}{n}\right) \]
- The area \( S_2 \) of the regular polygon circumscribing the circle is given by:
\[ S_2 = \frac{1}{2} n A^2 \tan\left(\frac{\pi}{n}\right) \]

2. **Setting Up the Ratio**:
- According to the problem, the ratio of the areas is:
\[ \frac{S_1}{S_2} = \frac{3}{4} \]
- Substituting the expressions for \( S_1 \) and \( S_2 \):
\[ \frac{\frac{1}{2} n A^2 \sin\left(\frac{2\pi}{n}\right)}{\frac{1}{2} n A^2 \tan\left(\frac{\pi}{n}\right)} = \frac{3}{4} \]
- This simplifies to:
\[ \frac{\sin\left(\frac{2\pi}{n}\right)}{\tan\left(\frac{\pi}{n}\right)} = \frac{3}{4} \]

3. **Using the Identity for Tangent**:
- Recall that \( \tan\left(\frac{\pi}{n}\right) = \frac{\sin\left(\frac{\pi}{n}\right)}{\cos\left(\frac{\pi}{n}\right)} \).
- Thus, we can rewrite the equation as:
\[ \frac{\sin\left(\frac{2\pi}{n}\right) \cos\left(\frac{\pi}{n}\right)}{\sin\left(\frac{\pi}{n}\right)} = \frac{3}{4} \]

4. **Using the Double Angle Identity**:
- Using the identity \( \sin(2x) = 2 \sin(x) \cos(x) \), we have:
\[ \frac{2 \sin\left(\frac{\pi}{n}\right) \cos\left(\frac{\pi}{n}\right) \cos\left(\frac{\pi}{n}\right)}{\sin\left(\frac{\pi}{n}\right)} = \frac{3}{4} \]
- This simplifies to:
\[ 2 \cos^2\left(\frac{\pi}{n}\right) = \frac{3}{4} \]

5. **Solving for Cosine**:
- Rearranging gives:
\[ \cos^2\left(\frac{\pi}{n}\right) = \frac{3}{8} \]
- Taking the square root:
\[ \cos\left(\frac{\pi}{n}\right) = \frac{\sqrt{3}}{2} \]

6. **Finding \( n \)**:
- The cosine value \( \frac{\sqrt{3}}{2} \) corresponds to angles:
\[ \frac{\pi}{n} = \frac{\pi}{6} \quad \text{or} \quad \frac{\pi}{n} = \frac{11\pi}{6} \]
- From \( \frac{\pi}{n} = \frac{\pi}{6} \), we find:
\[ n = 6 \]

### Conclusion:
Thus, the value of \( n \) is \( 6 \).