Question

If the area of a circle is equal to the area of a square, then the ratio of their perimeters is

• A. 1 : 1
• B. 2 : \( \pi \)
• C. \( \pi \) : 2
• D. \( \sqrt{\pi} : 2 \)

Hint:

Always verify the units and dimensions when dealing with geometric properties and relationships to ensure accurate calculations.

Answer 1

The Correct Option is D

Step 1: The area of the circle is \( \pi r^2 \). Let the side of the square be \( s \), then the area of the square is \( s^2 \). Step 2: Given that the areas of the circle and the square are equal: \[ \pi r^2 = s^2 \] Step 3: The perimeter of the circle is \( 2\pi r \) and the perimeter of the square is \( 4s \). Step 4: From \( \pi r^2 = s^2 \), we get \( s = \sqrt{\pi}r \). Step 5: The ratio of the perimeters is: \[ \frac{2\pi r}{4\sqrt{\pi} r} = \frac{\sqrt{\pi}}{2} \] Thus, the correct answer is \( \boxed{\frac{\sqrt{\pi}}{2}} \).