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

Options

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

Explanation

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

### Step 1: Understand the Perimeter Formulas
The perimeter (circumference) of a circle is given by the formula:
\[ C_{circle} = 2\pi r \]
where \( r \) is the radius of the circle.

The perimeter of a square is given by the formula:
\[ C_{square} = 4a \]
where \( a \) is the length of a side of the square.

### Step 2: Set the Perimeters Equal
Since the problem states that the perimeter of the circle is equal to the perimeter of the square, we can write:
\[ 2\pi r = 4a \]

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

### Step 4: Calculate the Areas
Now, we can calculate the areas of both shapes.

The area of the circle is given by:
\[ A_{circle} = \pi r^2 \]
Substituting \( r \) from Step 3:
\[ A_{circle} = \pi \left(\frac{2a}{\pi}\right)^2 = \pi \cdot \frac{4a^2}{\pi^2} = \frac{4a^2}{\pi} \]

The area of the square is given by:
\[ A_{square} = a^2 \]

### Step 5: Find the Ratio of the Areas
Now, we can find the ratio of the area of the circle to the area of the square:
\[ \text{Ratio} = \frac{A_{circle}}{A_{square}} = \frac{\frac{4a^2}{\pi}}{a^2} = \frac{4}{\pi} \]

### Step 6: Finalize the Ratio
Thus, the ratio of the areas of the circle to the square is:
\[ \text{Ratio} = \frac{4}{\pi} \]

### Conclusion
The final answer is that the ratio of the areas of the circle to the square is \( \frac{4}{\pi} \).

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