If the circumference of a circle is eqaul to the perimeter of a square, then what will be the ratio of their areas ? (Where `pi=22/7`)

Options

A. `14 : 11`
B. `21 : 1`
C. `11 : 4`
D. `5 : 7`

Answer

To solve the problem of finding the ratio of the areas of a circle and a square when their circumference and perimeter are equal, we can follow these steps:

### Step 1: Define the formulas
- The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

- The perimeter \( P \) of a square with side length \( a \) is given by:
\[ P = 4a \]

### Step 2: Set the circumference equal to the perimeter
Since the circumference of the circle is equal to the perimeter of the square, we can set the two equations equal to each other:
\[ 2\pi r = 4a \]

### Step 3: Solve for the radius \( r \)
To find \( r \), we can rearrange the equation:
\[ r = \frac{4a}{2\pi} = \frac{2a}{\pi} \]

### Step 4: Find the area of the circle
The area \( A_c \) of the circle is given by:
\[ A_c = \pi r^2 \]
Substituting the value of \( r \):
\[ A_c = \pi \left(\frac{2a}{\pi}\right)^2 = \pi \cdot \frac{4a^2}{\pi^2} = \frac{4a^2}{\pi} \]

### Step 5: Find the area of the square
The area \( A_s \) of the square is given by:
\[ A_s = a^2 \]

### Step 6: Find the ratio of the areas
Now, we can find the ratio of the area of the circle to the area of the square:
\[ \text{Ratio} = \frac{A_c}{A_s} = \frac{\frac{4a^2}{\pi}}{a^2} = \frac{4}{\pi} \]

### Step 7: Substitute the value of \( \pi \)
Given \( \pi = \frac{22}{7} \), we substitute this value into the ratio:
\[ \text{Ratio} = \frac{4}{\frac{22}{7}} = 4 \times \frac{7}{22} = \frac{28}{22} = \frac{14}{11} \]

### Final Answer
Thus, the ratio of the areas of the circle to the square is:
\[ \boxed{\frac{14}{11}} \]