The area of the square that can be inscribed in a circle of radius 8 cm is

Options

A. 256 `cm^(2)`
B. 128 `cm^(2)`
C. `sqrt(2)r^(2)`
D. 64 `cm^(2)`

Explanation

To find the area of the square that can be inscribed in a circle of radius 8 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the relationship between the circle and the inscribed square**:
- The diameter of the circle is equal to the diagonal of the square.

2. **Calculate the diameter of the circle**:
- The radius of the circle is given as 8 cm.
- Therefore, the diameter \( D \) can be calculated as:
\[ D = 2 \times \text{radius} = 2 \times 8 = 16 \text{ cm} \]

3. **Use the Pythagorean theorem**:
- In a square, if \( A \) is the length of a side, the diagonal \( d \) can be expressed using the Pythagorean theorem:
\[ d = A\sqrt{2} \]
- Since we know the diagonal is equal to the diameter of the circle, we can set up the equation:
\[ A\sqrt{2} = 16 \]

4. **Solve for the side length \( A \)**:
- Rearranging the equation gives:
\[ A = \frac{16}{\sqrt{2}} = 16 \times \frac{\sqrt{2}}{2} = 8\sqrt{2} \text{ cm} \]

5. **Calculate the area of the square**:
- The area \( A_{\text{square}} \) of the square is given by:
\[ A_{\text{square}} = A^2 = (8\sqrt{2})^2 = 64 \times 2 = 128 \text{ cm}^2 \]

### Final Answer:
The area of the square that can be inscribed in a circle of radius 8 cm is \( 128 \text{ cm}^2 \).

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