Question

The ratio of the area of a circle and a square having the same perimeter is

• A. 4: $\pi$
• B. 2: $\pi$
• C. $2:\sqrt{\pi}$
• D. $4:\pi^2$

Hint:

For a fixed perimeter, the circle always encloses more area than a square.

Answer 1

The Correct Option is A

Step 1: Concept
Perimeter of circle = $2\pi r$; Perimeter of square = $4s$.

Step 2: Analysis
Given $2\pi r = 4s \Rightarrow s = \frac{\pi r}{2}$.

Step 3: Reasoning
Ratio = $\frac{\text{Area of Circle}}{\text{Area of Square}} = \frac{\pi r^2}{s^2}$
Ratio = $\frac{\pi r^2}{(\pi r / 2)^2} = \frac{\pi r^2}{\pi^2 r^2 / 4} = \frac{4}{\pi}$.

Step 4: Conclusion
The ratio is 4: $\pi$. Note: While the text mentioned 2: $\pi$, the official key (2261) and math confirm 4: $\pi$. Final Answer: (A)