A square and a circle have equal perimeters. The ratio of the area of the square to the area of the circle is

Options

A. `1 : 1`
B. ` 1 : 4 `
C. ` pi : 2 `
D. ` pi : 4 `

Answer

To solve the problem where a square and a circle have equal perimeters and we need to find the ratio of the area of the square to the area of the circle, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Perimeters**:
- The perimeter of the square (P_square) is given by the formula:
\[ P_{square} = 4 \times A \]
where \( A \) is the length of a side of the square.
- The perimeter of the circle (P_circle) is given by the formula:
\[ P_{circle} = 2\pi r \]
where \( r \) is the radius of the circle.

2. **Set the Perimeters Equal**:
Since the perimeters are equal, we can set the two equations equal to each other:
\[ 4A = 2\pi r \]

3. **Solve for the Ratio \( \frac{A}{r} \)**:
To find the ratio of the side of the square to the radius of the circle, we can rearrange the equation:
\[ \frac{A}{r} = \frac{\pi}{2} \]

4. **Calculate the Areas**:
- The area of the square (Area_square) is given by:
\[ Area_{square} = A^2 \]
- The area of the circle (Area_circle) is given by:
\[ Area_{circle} = \pi r^2 \]

5. **Find the Ratio of Areas**:
We need to find the ratio of the area of the square to the area of the circle:
\[ \frac{Area_{square}}{Area_{circle}} = \frac{A^2}{\pi r^2} \]

6. **Substitute \( \frac{A}{r} \)**:
We can substitute \( \frac{A}{r} = \frac{\pi}{2} \) into the area ratio:
\[ \frac{Area_{square}}{Area_{circle}} = \frac{A^2}{\pi r^2} = \frac{(A/r)^2 \cdot r^2}{\pi r^2} = \frac{(\frac{\pi}{2})^2 \cdot r^2}{\pi r^2} \]

7. **Simplify the Ratio**:
The \( r^2 \) terms cancel out:
\[ \frac{Area_{square}}{Area_{circle}} = \frac{\left(\frac{\pi}{2}\right)^2}{\pi} = \frac{\frac{\pi^2}{4}}{\pi} = \frac{\pi}{4} \]

### Final Answer:
The ratio of the area of the square to the area of the circle is:
\[ \frac{\pi}{4} \]