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, we need to find the ratio of the area of a square to the area of a circle, given that they have equal perimeters.

### Step-by-Step Solution:

1. **Define Variables:**
- Let the side of the square be \( A \).
- Let the radius of the circle be \( R \).

2. **Write the Perimeter Formulas:**
- The perimeter of the square is given by:
\[ P_{\text{square}} = 4A \]
- The perimeter of the circle is given by:
\[ P_{\text{circle}} = 2\pi R \]

3. **Set the Perimeters Equal:**
Since the perimeters are equal, we can write:
\[ 4A = 2\pi R \]

4. **Solve for \( A \):**
Rearranging the equation to solve for \( A \):
\[ A = \frac{2\pi R}{4} = \frac{\pi R}{2} \]

5. **Calculate the Areas:**
- The area of the square is:
\[ \text{Area}_{\text{square}} = A^2 = \left(\frac{\pi R}{2}\right)^2 = \frac{\pi^2 R^2}{4} \]
- The area of the circle is:
\[ \text{Area}_{\text{circle}} = \pi R^2 \]

6. **Find the Ratio of Areas:**
Now, we can find the ratio of the area of the square to the area of the circle:
\[ \text{Ratio} = \frac{\text{Area}_{\text{square}}}{\text{Area}_{\text{circle}}} = \frac{\frac{\pi^2 R^2}{4}}{\pi R^2} \]

7. **Simplify the Ratio:**
Simplifying the ratio:
\[ \text{Ratio} = \frac{\pi^2 R^2}{4 \cdot \pi R^2} = \frac{\pi}{4} \]

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