Question

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

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

Hint:

For any given perimeter, a circle always encloses the maximum area compared to any polygon.
This means the ratio of circle area to square area must be greater than 1.
Among the options, only (A) 22:7 and (D) 14:11 are greater than 1.
Since 22:7 is much too large, 14:11 is the correct geometric ratio.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question is from Mensuration, dealing with the comparison of areas and boundaries of different shapes (circle and square).
We are given that their boundaries are equal, and we need to find the ratio of their areas.

Step 2: Key Formula or Approach:
Let the radius of the circle be \(r\), and the side length of the square be \(a\).
1. Perimeter of the circle (circumference) is:
\[ C = 2\pi r \]
2. Perimeter of the square is:
\[ P = 4a \]
Given that \(C = P\), we can express \(a\) in terms of \(r\).
Then, we compute the ratio of their areas:
\[ \text{Ratio} = \frac{\text{Area of Circle}}{\text{Area of Square}} = \frac{\pi r^2}{a^2} \]

Step 3: Detailed Explanation:
Set the perimeters equal:
\[ 2\pi r = 4a \]
Simplify to express \(a\) in terms of \(r\):
\[ a = \frac{2\pi r}{4} = \frac{\pi r}{2} \]
Now, write the formula for the areas:
- Area of the circle:
\[ A_{\text{circle}} = \pi r^2 \]
- Area of the square:
\[ A_{\text{square}} = a^2 \]
Substitute \(a = \frac{\pi r}{2}\) into the square's area formula:
\[ A_{\text{square}} = \left(\frac{\pi r}{2}\right)^2 = \frac{\pi^2 r^2}{4} \]
Calculate the ratio of the areas:
\[ \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi r^2}{\frac{\pi^2 r^2}{4}} \]
Cancel \(r^2\) and simplify the fraction:
\[ \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi}{\frac{\pi^2}{4}} = \frac{4}{\pi} \] Substitute \(\pi = \frac{22}{7}\):
\[ \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{4}{\frac{22}{7}} = \frac{4 \times 7}{22} \]
\[ \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{28}{22} = \frac{14}{11} \]

Step 4: Final Answer:
The ratio of their areas is 14 : 11.