If the perimeter of a circle is equal to that of a square, then the ratio of their areas is

Options

A. `22:7`
B. `14:11`
C. `7:22`
D. `11:14`

Explanation

To solve the problem step by step, we will find the ratio of the areas of a circle and a square when their perimeters are equal.

### Step-by-Step Solution:

**Step 1: Understand the Perimeters**
- The perimeter of a circle (circumference) is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

- The perimeter of a square is given by the formula:
\[ P = 4a \]
where \( a \) is the length of a side of the square.

**Step 2: Set the Perimeters Equal**
- According to the problem, the perimeter of the circle is equal to the perimeter of the square:
\[ 2\pi r = 4a \]

**Step 3: Solve for the Radius \( r \)**
- Rearranging the equation to find \( r \):
\[ r = \frac{4a}{2\pi} = \frac{2a}{\pi} \]

**Step 4: Find the Areas**
- The area of the circle is given by:
\[ A_{circle} = \pi r^2 \]
Substituting \( r = \frac{2a}{\pi} \):
\[ A_{circle} = \pi \left(\frac{2a}{\pi}\right)^2 = \pi \cdot \frac{4a^2}{\pi^2} = \frac{4a^2}{\pi} \]

- The area of the square is:
\[ A_{square} = a^2 \]

**Step 5: Calculate the Ratio of Areas**
- The ratio of the area of the circle to the area of the square is:
\[ \text{Ratio} = \frac{A_{circle}}{A_{square}} = \frac{\frac{4a^2}{\pi}}{a^2} \]
Simplifying this gives:
\[ \text{Ratio} = \frac{4}{\pi} \]

**Step 6: Express the Ratio in a Different Form**
- To express this ratio in terms of whole numbers, we can multiply both sides by \( \pi \):
\[ \text{Ratio} = \frac{4}{\pi} \Rightarrow 4 : \pi \]
- Using \( \pi \approx \frac{22}{7} \), we can convert this ratio:
\[ 4 : \frac{22}{7} \Rightarrow 4 \times 7 : 22 \Rightarrow 28 : 22 \]
- Simplifying this ratio by dividing both sides by 2:
\[ 14 : 11 \]

### Final Answer:
The ratio of the areas of the circle to the square is:
\[ \boxed{14 : 11} \]