Question

The perimeter of a square is P cm and the area of the circle inscribed in it is A \({cm}^2\). If P = 4A, then what is the length of a side of the square?

• A. \(\frac{4}{\pi} cm\)
• B. \(\frac{12}{15}\)
• C. \(\frac{14}{7}\)
• D. \(\frac{16}{\pi} cm\)

Answer 1

The Correct Option is A

Step 1: Let the side of the square be \( s \). Then, the perimeter of the square is: \[ P = 4s. \] Step 2: The area of the circle inscribed in the square is given by \( A = \pi r^2 \), where \( r \) is the radius of the circle. The radius of the circle is half the side of the square: \[ r = \frac{s}{2}. \] Thus, the area of the circle is: \[ A = \pi \left(\frac{s}{2}\right)^2 = \frac{\pi s^2}{4}. \] Step 3: We are given that \( P = 4A \). Substituting for \( P \) and \( A \): \[ 4s = 4 \times \frac{\pi s^2}{4}. \] Simplifying: \[ s = \pi s^2 \quad \Rightarrow \quad s = \frac{4}{\pi}. \] Conclusion: The length of the side of the square is \( \frac{4}{\pi} \, \text{cm} \).