If the circumference of a circle and the perimeter of a square are equal, then

Options

A. Area of the circle = Area of the square
B. Area of the circle `gt` Area of the square
C. Area of the circle `lt` Area of the square
D. Nothing definite can be said about the relationship between the areas of the circle and square

Explanation

To solve the problem step by step, we need to find the relationship between the area of a circle and the area of a square when their circumference and perimeter are equal.

### Step 1: Define the variables
Let the radius of the circle be \( R \) and the side length of the square be \( A \).

**Hint:** Remember that the circumference of a circle and the perimeter of a square are defined in terms of their dimensions.

### Step 2: Write the formulas for circumference and perimeter
The circumference \( C \) of the circle is given by:
\[ C = 2\pi R \]
The perimeter \( P \) of the square is given by:
\[ P = 4A \]

**Hint:** Use the formulas for circumference and perimeter to set up the equation based on the problem statement.

### Step 3: Set the circumference equal to the perimeter
According to the problem, the circumference of the circle is equal to the perimeter of the square:
\[ 2\pi R = 4A \]

**Hint:** This equation will help you express one variable in terms of the other.

### Step 4: Solve for \( R \) in terms of \( A \)
Rearranging the equation gives:
\[ R = \frac{4A}{2\pi} = \frac{2A}{\pi} \]

**Hint:** Simplifying the equation can help you find a relationship between the radius of the circle and the side of the square.

### Step 5: Calculate the area of the circle
The area \( A_c \) of the circle is given by:
\[ A_c = \pi R^2 \]
Substituting \( R = \frac{2A}{\pi} \):
\[ A_c = \pi \left(\frac{2A}{\pi}\right)^2 = \pi \cdot \frac{4A^2}{\pi^2} = \frac{4A^2}{\pi} \]

**Hint:** Make sure to square the radius correctly and simplify the expression.

### Step 6: Calculate the area of the square
The area \( A_s \) of the square is given by:
\[ A_s = A^2 \]

**Hint:** The area of the square is straightforward; just square the side length.

### Step 7: Compare the areas
Now we need to compare the area of the circle and the area of the square:
\[ A_c = \frac{4A^2}{\pi} \quad \text{and} \quad A_s = A^2 \]
To compare, we can express the relationship:
\[ \frac{A_c}{A_s} = \frac{\frac{4A^2}{\pi}}{A^2} = \frac{4}{\pi} \]

**Hint:** Knowing the value of \( \pi \) (approximately 3.14) can help you determine the relationship.

### Step 8: Determine the relationship
Since \( \frac{4}{\pi} > 1 \) (because \( 4 > 3.14 \)), we conclude that:
\[ A_c > A_s \]

**Hint:** This means the area of the circle is greater than the area of the square.

### Conclusion
Thus, the area of the circle is greater than the area of the square when their circumference and perimeter are equal.

**Final Answer:** The area of the circle is greater than the area of the square.