The areas of a circle and a square are same. The ratio of the side of the square to the radius of the circle is

Options

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

Answer

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

### Step-by-Step Solution:

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

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

3. **Set the Areas Equal:**
According to the problem, the areas of the square and the circle are the same:
\[ a^2 = \pi r^2 \]

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

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

6. **Express the Ratio:**
Therefore, the ratio of the side of the square to the radius of the circle is:
\[ \frac{a}{r} = \sqrt{\pi} \]

### Final Answer:
The ratio of the side of the square to the radius of the circle is \( \sqrt{\pi} : 1 \).

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