If the area of a circle inscribed in a square is `9pi 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 when the area of the inscribed circle is given, we can follow these steps:

### Step 1: Understand the relationship between the circle and the square
The circle is inscribed in the square, which means that the diameter of the circle is equal to the side length of the square.

### Step 2: Use the area of the circle to find the radius
The area of the circle is given as \(9\pi \, \text{cm}^2\). The formula for the area of a circle is:
\[ \text{Area} = \pi r^2 \]
where \(r\) is the radius of the circle.

### Step 3: Set up the equation
From the area of the circle, we can set up the equation:
\[ \pi r^2 = 9\pi \]

### Step 4: Solve for \(r^2\)
Dividing both sides by \(\pi\):
\[ r^2 = 9 \]

### Step 5: Find the radius \(r\)
Taking the square root of both sides:
\[ r = 3 \, \text{cm} \]

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

### Step 7: Find the area of the square
Since the diameter of the circle is equal to the side length \(s\) of the square, we have:
\[ s = d = 6 \, \text{cm} \]
Now, the area \(A\) of the square is given by:
\[ A = s^2 = 6^2 = 36 \, \text{cm}^2 \]

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

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