Question

If the perimeter of a circle is equal to that of a square, then find the ratio of their areas.

• A. $ 11:14 $
• B. $ 5:6 $
• C. $ 14:11 $
• D. $ 7:8 $

Hint:

When comparing the areas of a circle and a square with equal perimeters, use the relationship between their dimensions to derive the ratio of their areas. Approximating $ \pi $ as $ \frac{22}{7} $ can help simplify calculations and arrive at the correct integer ratio.

Answer 1

The Correct Option is C

Step 1: Define Variables and Equations.
Let the radius of the circle be \( r \) and the side length of the square be \( s \).
Perimeter of the circle: \( 2\pi r \).
Perimeter of the square: \( 4s \).
Given that the perimeters are equal: \[ 2\pi r = 4s. \] Solve for \( s \) in terms of \( r \): \[ s = \frac{\pi r}{2}. \] Step 2: Calculate the Areas.
Area of the circle: \( \pi r^2 \). Area of the square: \[ s^2 = \left(\frac{\pi r}{2}\right)^2 = \frac{\pi^2 r^2}{4}. \] Step 3: Find the Ratio of Areas.
The ratio of the area of the circle to the area of the square is: \[ \text{Ratio} = \frac{\text{Area of circle}}{\text{Area of square}} = \frac{\pi r^2}{\frac{\pi^2 r^2}{4}} = \frac{\pi r^2 \cdot 4}{\pi^2 r^2} = \frac{4}{\pi}. \] To express this ratio in terms of integers, approximate \( \pi \approx \frac{22}{7} \): \[ \frac{4}{\pi} \approx \frac{4}{\frac{22}{7}} = \frac{4 \times 7}{22} = \frac{28}{22} = \frac{14}{11}. \] Thus, the ratio of the areas is \( 14:11 \). Step 4: Analyze the Options.
Option (1): \( 11:14 \) — Incorrect, as this is the reciprocal of the correct ratio.
Option (2): \( 5:6 \) — Incorrect, as this does not match the calculated value.
Option (3): \( 14:11 \) — Correct, as it matches the calculated value.
Option (4): \( 7:8 \) — Incorrect, as this does not match the calculated value. Step 5: Final Answer. \[ (3) \quad \mathbf{14:11} \]