If the area of a circle and a square are equal, then the ratio of their perimeter is

Options

A. `1:1`
B. `2:pi`
C. `pi:2`
D. `sqrt(pi):2`

Answer

To solve the problem, we need to find the ratio of the perimeter of a circle and a square when their areas are equal.

### Step-by-Step Solution:

1. **Define the Variables**:
Let the radius of the circle be \( r \) and the side length of the square be \( a \).

2. **Write the Area Formulas**:
- The area of the circle is given by the formula:
\[ \text{Area of Circle} = \pi r^2 \]
- The area of the square is given by the formula:
\[ \text{Area of Square} = a^2 \]

3. **Set the Areas Equal**:
Since the areas are equal, we can set the two area formulas equal to each other:
\[ \pi r^2 = a^2 \]

4. **Rearrange to Find the Ratio of \( r \) and \( a \)**:
Rearranging the equation gives:
\[ \frac{r^2}{a^2} = \frac{1}{\pi} \]
Taking the square root of both sides, we find:
\[ \frac{r}{a} = \frac{1}{\sqrt{\pi}} \]

5. **Write the Perimeter Formulas**:
- The perimeter of the circle (circumference) is given by:
\[ \text{Perimeter of Circle} = 2\pi r \]
- The perimeter of the square is given by:
\[ \text{Perimeter of Square} = 4a \]

6. **Find the Ratio of the Perimeters**:
We need to find the ratio of the perimeter of the circle to the perimeter of the square:
\[ \text{Ratio} = \frac{\text{Perimeter of Circle}}{\text{Perimeter of Square}} = \frac{2\pi r}{4a} \]
Simplifying this gives:
\[ \text{Ratio} = \frac{\pi r}{2a} \]

7. **Substitute the Ratio of \( r \) and \( a \)**:
From step 4, we know that \( \frac{r}{a} = \frac{1}{\sqrt{\pi}} \). Substituting this into the ratio gives:
\[ \text{Ratio} = \frac{\pi \left(\frac{1}{\sqrt{\pi}} a\right)}{2a} \]
Simplifying this further:
\[ \text{Ratio} = \frac{\pi}{2\sqrt{\pi}} = \frac{\sqrt{\pi}}{2} \]

8. **Final Result**:
Thus, the ratio of the perimeter of the circle to the perimeter of the square is:
\[ \text{Ratio} = \frac{\sqrt{\pi}}{2} \]

### Conclusion:
The required ratio of the perimeter of the circle to the perimeter of the square is \( \sqrt{\pi} : 2 \).

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