If the area of a circle inscribed in a square is `9 pi cm^(2)` , then the area of the square is

Options

A. `24 cm^(2)`
B. `30 cm^(2)`
C. `36 cm^2`
D. `81 cm^2`

Answer

To find the area of the square in which a circle is inscribed, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between the circle and the square:**
The area of a circle inscribed in a square is given, and we know that the diameter of the circle is equal to the side length of the square.

2. **Use the formula for the area of a circle:**
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

3. **Set up the equation with the given area:**
We know that the area of the inscribed circle is \( 9\pi \, \text{cm}^2 \). Therefore, we can set up the equation:
\[ \pi r^2 = 9\pi \]

4. **Cancel \( \pi \) from both sides:**
Dividing both sides by \( \pi \) gives:
\[ r^2 = 9 \]

5. **Solve for the radius \( r \):**
Taking the square root of both sides, we find:
\[ r = \sqrt{9} = 3 \, \text{cm} \]

6. **Determine the diameter of the circle:**
The diameter \( d \) of the circle is twice the radius:
\[ d = 2r = 2 \times 3 = 6 \, \text{cm} \]

7. **Relate the diameter to the side of the square:**
Since the diameter of the circle is equal to the side length \( s \) of the square, we have:
\[ s = d = 6 \, \text{cm} \]

8. **Calculate the area of the square:**
The area \( A_s \) of the square is given by:
\[ A_s = s^2 = 6^2 = 36 \, \text{cm}^2 \]

### Final Answer:
The area of the square is \( 36 \, \text{cm}^2 \).

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