The area of a square is equal to the area of a circle. What is the ratio between the side of the square and the radius of the circle?

Options

A. `sqrtpi:1`
B. `1:sqrt3`
C. `1:pi`
D. `pi:1`

Answer

To solve the problem step by step, we need to find the ratio between the side of a square and the radius of a circle, given that their areas are equal.

### Step 1: Write the formulas for the areas
The area of a square (A_square) is given by:
\[ A_{\text{square}} = S^2 \]
where \( S \) is the side length of the square.

The area of a circle (A_circle) is given by:
\[ A_{\text{circle}} = \pi R^2 \]
where \( R \) is the radius of the circle.

### Step 2: Set the areas equal to each other
Since the problem states that the area of the square is equal to the area of the circle, we can write:
\[ S^2 = \pi R^2 \]

### Step 3: Rearrange the equation to find the ratio
To find the ratio \( \frac{S}{R} \), we can rearrange the equation:
\[ \frac{S^2}{R^2} = \pi \]

Taking the square root of both sides gives:
\[ \frac{S}{R} = \sqrt{\pi} \]

### Step 4: Write the ratio in the required form
Thus, the ratio of the side of the square to the radius of the circle can be expressed as:
\[ S : R = \sqrt{\pi} : 1 \]

### Final Answer
The ratio between the side of the square and the radius of the circle is:
\[ S : R = \sqrt{\pi} : 1 \]

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