A copper wire when bent in the form of a square encloses an area of `121 cm^(2)`. Y the same wire is bent in the form of a circle, find the area enclosed by it.

Options

A. `184cm^(2)`
B. `164cm^(2)`
C. `154cm^(2)`
D. `174cm^(2)`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the area of the square
The area of the square is given as \(121 \, \text{cm}^2\). The formula for the area of a square is:
\[ \text{Area} = a^2 \]
where \(a\) is the length of one side of the square.

### Step 2: Find the side length of the square
To find the side length \(a\), we take the square root of the area:
\[ a = \sqrt{121} = 11 \, \text{cm} \]

### Step 3: Calculate the perimeter of the square
The perimeter \(P\) of a square can be calculated using the formula:
\[ P = 4a \]
Substituting the value of \(a\):
\[ P = 4 \times 11 = 44 \, \text{cm} \]

### Step 4: Relate the perimeter of the square to the circumference of the circle
When the wire is bent into the shape of a circle, the perimeter of the square is equal to the circumference \(C\) of the circle:
\[ C = P = 44 \, \text{cm} \]

### Step 5: Use the circumference to find the radius of the circle
The formula for the circumference of a circle is:
\[ C = 2\pi r \]
Setting this equal to the perimeter we found:
\[ 44 = 2\pi r \]
Now, we can solve for \(r\):
\[ r = \frac{44}{2\pi} = \frac{22}{\pi} \]
Using \(\pi \approx \frac{22}{7}\):
\[ r = \frac{22 \times 7}{22} = 7 \, \text{cm} \]

### Step 6: Calculate the area of the circle
The area \(A\) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of \(r\):
\[ A = \pi (7^2) = \pi \times 49 \]
Using \(\pi \approx \frac{22}{7}\):
\[ A = \frac{22}{7} \times 49 = 22 \times 7 = 154 \, \text{cm}^2 \]

### Final Answer
The area enclosed by the wire when bent in the form of a circle is:
\[ \boxed{154 \, \text{cm}^2} \]

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