The perimeter of a circle is equal to the perimeter of a square. Then.their area are in the ratio (use `pi = 22/7`)

Options

A. `4:1`
B. `3:7`
C. `14:11`
D. `22:7`

Answer

To solve the problem where the perimeter of a circle is equal to the perimeter of a square, and we need to find the ratio of their areas, we can follow these steps:

### Step 1: Understand the Perimeters
The perimeter (circumference) of a circle 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:
\[ P = 4a \]
where \( a \) is the length of one 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: Simplify the Equation
We can simplify this equation to find a relationship between \( r \) and \( a \):
\[ \pi r = 2a \]
\[ r = \frac{2a}{\pi} \]

### Step 4: Calculate the Areas
Now we need to find the areas of both shapes.

The area of the circle is given by:
\[ A_{circle} = \pi r^2 \]

Substituting \( r \) from the previous step:
\[ A_{circle} = \pi \left(\frac{2a}{\pi}\right)^2 \]
\[ A_{circle} = \pi \cdot \frac{4a^2}{\pi^2} \]
\[ A_{circle} = \frac{4a^2}{\pi} \]

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

### Step 5: Find the Ratio of Areas
Now we can find the ratio of the area of the circle to the area of the square:
\[ \text{Ratio} = \frac{A_{circle}}{A_{square}} = \frac{\frac{4a^2}{\pi}}{a^2} \]
\[ \text{Ratio} = \frac{4}{\pi} \]

### Step 6: Substitute the Value of \(\pi\)
Using the value of \(\pi = \frac{22}{7}\):
\[ \text{Ratio} = \frac{4}{\frac{22}{7}} = 4 \cdot \frac{7}{22} = \frac{28}{22} = \frac{14}{11} \]

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

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