If the circumference of a circle is equal to the perimeter of square, then which one of the following is correct?

Options

A. Area of circle = Area of square
B. Area of circle `gt` Area of square
C. Area of circle `lt` Area of square
D. None of the above

Answer

To solve the problem, we need to analyze the relationship between the circumference of a circle and the perimeter of a square. Let's break it down step by step.

### Step 1: Understand the formulas
- The **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 the formula:
\[ P = 4a \]
where \( a \) is the length of a side of the square.

### Step 2: Set the two equations equal
According to the problem, the circumference of the circle is equal to the perimeter of the square:
\[ 2\pi r = 4a \]

### Step 3: Simplify the equation
We can simplify the equation by dividing both sides by 2:
\[ \pi r = 2a \]
Now, we can express \( r \) in terms of \( a \):
\[ r = \frac{2a}{\pi} \]

### Step 4: Calculate the areas
Next, we will calculate the areas of both the circle and the square.

- 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 given by:
\[ A_{square} = a^2 \]

### Step 5: Compare the areas
Now, we need to compare \( A_{circle} \) and \( A_{square} \):
\[ A_{circle} = \frac{4a^2}{\pi} \quad \text{and} \quad A_{square} = a^2 \]
To compare, we can look at the ratio:
\[ \frac{A_{circle}}{A_{square}} = \frac{\frac{4a^2}{\pi}}{a^2} = \frac{4}{\pi} \]
Since \( \pi \) is approximately 3.14, we have:
\[ \frac{4}{\pi} > 1 \quad \text{(because 4 > 3.14)} \]
This means:
\[ A_{circle} > A_{square} \]

### Conclusion
Thus, the area of the circle is greater than the area of the square.

### Final Answer
The correct option is that the area of the circle is greater than the area of the square.

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