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

Options

A. `1 : pi`
B. `pi :1`
C. `pi :2`
D. `2 :pi`

Answer

To find the ratio of the areas of a circle and a square where the radius of the circle is equal to the side of the square, we can follow these steps:

### Step 1: Define the variables
Let the radius of the circle be \( r \). Since the radius of the circle is also the side of the square, the side length of the square is also \( r \).

### Step 2: Calculate the area of the circle
The formula for the area of a circle is given by:
\[ \text{Area of Circle} = \pi r^2 \]

### Step 3: Calculate the area of the square
The formula for the area of a square is given by:
\[ \text{Area of Square} = \text{side}^2 = r^2 \]

### Step 4: Set up the ratio of the areas
Now, we need to find the ratio of the area of the circle to the area of the square:
\[ \text{Ratio} = \frac{\text{Area of Circle}}{\text{Area of Square}} = \frac{\pi r^2}{r^2} \]

### Step 5: Simplify the ratio
When we simplify the ratio, we can cancel \( r^2 \) from the numerator and the denominator:
\[ \text{Ratio} = \pi \]

### Final Answer
The ratio of the areas of the circle to the square is \( \pi : 1 \).

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