A wire, when bent in the form of a square, encloses an area of 196 `cm^(2)`. If the same wire is bent to form a circle, find the area of the circle.

Answer

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

### Step 1: Find the side of the square
Given the area of the square is 196 cm², we can use the formula for the area of a square:
\[ \text{Area} = \text{side}^2 \]
Let the side of the square be \( x \):
\[ x^2 = 196 \]
To find \( x \), we take the square root of both sides:
\[ x = \sqrt{196} \]

### Step 2: Calculate the square root of 196
Finding the square root of 196:
\[ \sqrt{196} = 14 \quad (\text{since } 14 \times 14 = 196) \]

### Step 3: Find the perimeter of the square
The perimeter \( P \) of a square is given by:
\[ P = 4 \times \text{side} \]
Substituting the side we found:
\[ P = 4 \times 14 = 56 \text{ cm} \]

### Step 4: Use the perimeter to find the circumference of the circle
Since the wire is bent to form a circle, the circumference \( C \) of the circle is equal to the perimeter of the square:
\[ C = 56 \text{ cm} \]

### Step 5: Find the radius of the circle
The formula for the circumference of a circle is:
\[ C = 2\pi r \]
Setting the circumference equal to 56 cm:
\[ 56 = 2\pi r \]
To find the radius \( r \), we rearrange the equation:
\[ r = \frac{56}{2\pi} = \frac{28}{\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{28 \times 7}{22} = \frac{196}{22} = \frac{98}{11} \text{ cm} \]

### Step 6: Find the area of the circle
The area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
Substituting the radius we found:
\[ A = \pi \left(\frac{98}{11}\right)^2 \]
Calculating \( r^2 \):
\[ \left(\frac{98}{11}\right)^2 = \frac{9604}{121} \]
Thus, the area becomes:
\[ A = \pi \times \frac{9604}{121} \]
Using \( \pi \approx \frac{22}{7} \):
\[ A = \frac{22}{7} \times \frac{9604}{121} \]
Calculating:
\[ A = \frac{22 \times 9604}{7 \times 121} = \frac{211328}{847} \]
Now, performing the division:
\[ A \approx 249.4 \text{ cm}^2 \]

### Final Answer
The area of the circle is approximately \( 249.4 \text{ cm}^2 \).

---