The area of a circle is same as the areas of a square. What is the ratio of the diameter of the circle and diagonal of the square ?

Options

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

Answer

To find the ratio of the diameter of a circle to the diagonal of a square when their areas are equal, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Areas**:
- The area of a circle is given by the formula:
\[ \text{Area}_{\text{circle}} = \pi r^2 \]
- The area of a square is given by the formula:
\[ \text{Area}_{\text{square}} = a^2 \]
where \( r \) is the radius of the circle and \( a \) is the side length of the square.

2. **Set the Areas Equal**:
Since the areas are equal, we have:
\[ \pi r^2 = a^2 \]

3. **Rearrange the Equation**:
To find the relationship between \( r \) and \( a \), we can rearrange the equation:
\[ \frac{r^2}{a^2} = \frac{1}{\pi} \]

4. **Take the Square Root**:
Taking the square root of both sides gives:
\[ \frac{r}{a} = \frac{1}{\sqrt{\pi}} \]

5. **Express the Diameter and Diagonal**:
- The diameter \( d \) of the circle is:
\[ d = 2r \]
- The diagonal \( D \) of the square is:
\[ D = a\sqrt{2} \]

6. **Form the Ratio**:
We need to find the ratio \( \frac{d}{D} \):
\[ \frac{d}{D} = \frac{2r}{a\sqrt{2}} \]

7. **Substitute \( r \) in terms of \( a \)**:
From the earlier step, we can express \( r \) in terms of \( a \):
\[ r = \frac{a}{\sqrt{\pi}} \]
Substituting this into the ratio gives:
\[ \frac{d}{D} = \frac{2 \left(\frac{a}{\sqrt{\pi}}\right)}{a\sqrt{2}} = \frac{2}{\sqrt{\pi}\sqrt{2}} = \frac{2}{\sqrt{2\pi}} \]

8. **Simplify the Ratio**:
We can further simplify this:
\[ \frac{d}{D} = \frac{2\sqrt{2}}{2\sqrt{\pi}} = \frac{\sqrt{2}}{\sqrt{\pi}} \]

### Final Answer:
The ratio of the diameter of the circle to the diagonal of the square is:
\[ \frac{\sqrt{2}}{\sqrt{\pi}} \]