Area of the circle inscribed in a square of diagonal `6 sqrt2cm ` (in sq. `cm` ) is

Options

A. `9pi`
B. `6pi`
C. `3pi`
D. `9sqrt2`

Answer

To solve the problem of finding the area of the circle inscribed in a square with a diagonal of \(6\sqrt{2}\) cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between the diagonal and the side of the square**:
- The diagonal \(d\) of a square is related to its side length \(s\) by the formula:
\[ d = s\sqrt{2} \]
- Given that the diagonal \(d = 6\sqrt{2}\) cm, we can set up the equation:
\[ 6\sqrt{2} = s\sqrt{2} \]

2. **Solve for the side length \(s\)**:
- To find \(s\), divide both sides of the equation by \(\sqrt{2}\):
\[ s = \frac{6\sqrt{2}}{\sqrt{2}} = 6 \text{ cm} \]

3. **Determine the diameter of the inscribed circle**:
- The diameter of the inscribed circle is equal to the side length of the square. Therefore:
\[ \text{Diameter} = s = 6 \text{ cm} \]

4. **Calculate the radius of the circle**:
- The radius \(r\) of the circle is half of the diameter:
\[ r = \frac{6}{2} = 3 \text{ cm} \]

5. **Find the area of the circle**:
- The area \(A\) of a circle is given by the formula:
\[ A = \pi r^2 \]
- Substituting the value of \(r\):
\[ A = \pi (3)^2 = 9\pi \text{ cm}^2 \]

### Final Answer:
The area of the circle inscribed in the square is \(9\pi\) cm².

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