What is the area of the square ?
I. The length of the diagonal of the square is 60 cm.
II. The perimeter of the square is equal to the circumference of the circle.

Options

A. A. If statement I alone is sufficient but statement II alone is not sufficient.
B. B. If statement Il alone is sufficient but statement I alone is not sufficient.
C. C. If each statement alone (either I or II) is sufficient.
D. D. If statement I and Il together are not sufficient

Answer

To find the area of the square based on the given statements, we can analyze each statement step by step.

### Step 1: Analyze Statement I
**Statement I:** The length of the diagonal of the square is 60 cm.

1. **Understanding the relationship between the diagonal and the side of the square:**
The diagonal (D) of a square can be related to the side length (a) using the formula:
\[ D = a\sqrt{2} \]

2. **Substituting the given diagonal length:**
Given that \( D = 60 \) cm, we can set up the equation:
\[ 60 = a\sqrt{2} \]

3. **Solving for the side length (a):**
Rearranging the equation gives:
\[ a = \frac{60}{\sqrt{2}} = 30\sqrt{2} \text{ cm} \]

4. **Calculating the area of the square:**
The area (A) of a square is given by:
\[ A = a^2 \]
Substituting the value of \( a \):
\[ A = (30\sqrt{2})^2 = 30^2 \times 2 = 900 \times 2 = 1800 \text{ cm}^2 \]

### Conclusion from Statement I:
From Statement I alone, we can determine that the area of the square is \( 1800 \text{ cm}^2 \).

### Step 2: Analyze Statement II
**Statement II:** The perimeter of the square is equal to the circumference of the circle.

1. **Understanding the perimeter of the square:**
The perimeter (P) of a square is given by:
\[ P = 4a \]

2. **Understanding the circumference of the circle:**
The circumference (C) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

3. **Setting the perimeter equal to the circumference:**
From Statement II, we have:
\[ 4a = 2\pi r \]
Rearranging gives:
\[ a = \frac{\pi r}{2} \]

4. **Insufficient information:**
Without knowing either the radius \( r \) or the side length \( a \), we cannot find the area of the square. Thus, Statement II alone is not sufficient to determine the area.

### Final Conclusion:
- **Statement I alone is sufficient** to find the area of the square.
- **Statement II alone is not sufficient** to find the area of the square.

### Final Answer:
The correct option is A: Statement I alone is sufficient, but Statement II alone is not sufficient.