A circle with area 616 `cm^2`' is bent and made into a square. What is the area of the square formed?

Options

A. `196cm^2`
B. `441cm^2`
C. `144cm^2`
D. `484cm^2`

Answer

To find the area of the square formed from a circle with an area of 616 cm², we can follow these steps:

### Step 1: Find the radius of the circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Given that the area of the circle is 616 cm², we can set up the equation:
\[ \pi r^2 = 616 \]
Using \( \pi \approx \frac{22}{7} \), we can rewrite the equation as:
\[ \frac{22}{7} r^2 = 616 \]

### Step 2: Solve for \( r^2 \)
To isolate \( r^2 \), multiply both sides by \( \frac{7}{22} \):
\[ r^2 = 616 \times \frac{7}{22} \]
Calculating \( 616 \div 22 \):
\[ 616 \div 22 = 28 \]
Now, multiply by 7:
\[ r^2 = 28 \times 7 = 196 \]

### Step 3: Find the radius \( r \)
Taking the square root of both sides gives us:
\[ r = \sqrt{196} = 14 \text{ cm} \]

### Step 4: Find the circumference of the circle
The circumference \( C \) of a circle is given by:
\[ C = 2 \pi r \]
Substituting the value of \( r \):
\[ C = 2 \times \frac{22}{7} \times 14 \]
Calculating this:
\[ C = 2 \times 22 \times 2 = 88 \text{ cm} \]

### Step 5: Relate the circumference to the perimeter of the square
The circumference of the circle is equal to the perimeter \( P \) of the square:
\[ P = 4 \times \text{side} \]
Setting the two equal gives:
\[ 88 = 4 \times \text{side} \]

### Step 6: Solve for the side of the square
Dividing both sides by 4:
\[ \text{side} = \frac{88}{4} = 22 \text{ cm} \]

### Step 7: Find the area of the square
The area \( A_s \) of the square is given by:
\[ A_s = \text{side}^2 = 22^2 = 484 \text{ cm}^2 \]

### Final Answer
Thus, the area of the square formed is:
\[ \boxed{484 \text{ cm}^2} \]