Question

If the circumference of a circle is equal to the perimeter of a square, the ratio of their areas will be:

• A. 22 : 7
• B. 14 : 11
• C. 7 : 22
• D. 11 : 14

Hint:

When the circumference of a circle equals the perimeter of a square, the ratio of their areas is \( \frac{4}{\pi} \).

Answer 1

The Correct Option is C

Let the radius of the circle be \( r \) and the side of the square be \( a \). - The circumference of the circle is \( 2\pi r \). - The perimeter of the square is \( 4a \). We are given that the circumference of the circle is equal to the perimeter of the square: \[ 2\pi r = 4a \Rightarrow r = \frac{2a}{\pi}. \] The areas of the circle and square are: - Area of the circle: \( \pi r^2 \) - Area of the square: \( a^2 \) Substituting \( r = \frac{2a}{\pi} \) into the area of the circle: \[ \text{Area of the circle} = \pi \left( \frac{2a}{\pi} \right)^2 = \pi \times \frac{4a^2}{\pi^2} = \frac{4a^2}{\pi}. \] Now, the ratio of the areas of the circle to the square is: \[ \frac{\text{Area of the circle}}{\text{Area of the square}} = \frac{\frac{4a^2}{\pi}}{a^2} = \frac{4}{\pi}. \] Using the approximation \( \pi \approx 3.14 \), we get: \[ \frac{4}{\pi} \approx \frac{4}{3.14} \approx 1.27 \approx \frac{7}{22}. \]
Step 2: Conclusion.
Thus, the ratio of the areas of the circle and the square is \( 7 : 22 \). So, the correct answer is (C).