A square and a circle are formed using pieces of wire of length 5024cm each . The ratio of the area of the square to that of the circle is

Options

A. `4 : pi`
B. ` pi : 8`
C. ` 8 : pi`
D. `pi : 4`

Answer

To find the ratio of the area of a square to that of a circle formed using pieces of wire of length 5024 cm each, we can follow these steps:

### Step 1: Determine the Perimeter of the Square
The length of the wire used to form the square is 5024 cm, which is also the perimeter of the square.

\[ \text{Perimeter of the square} = 4s \]

Where \( s \) is the length of one side of the square.

### Step 2: Calculate the Side Length of the Square
To find the side length \( s \), we can rearrange the perimeter formula:

\[ s = \frac{\text{Perimeter}}{4} = \frac{5024}{4} = 1256 \text{ cm} \]

### Step 3: Calculate the Area of the Square
The area \( A_s \) of the square is given by:

\[ A_s = s^2 = (1256)^2 \]

Calculating \( 1256^2 \):

\[ A_s = 1576336 \text{ cm}^2 \]

### Step 4: Determine the Circumference of the Circle
The length of the wire used to form the circle is also 5024 cm, which is the circumference of the circle.

\[ \text{Circumference of the circle} = 2\pi r \]

Where \( r \) is the radius of the circle.

### Step 5: Calculate the Radius of the Circle
To find the radius \( r \), we can rearrange the circumference formula:

\[ r = \frac{\text{Circumference}}{2\pi} = \frac{5024}{2\pi} = \frac{5024}{2 \times 3.14} \approx 799.37 \text{ cm} \]

### Step 6: Calculate the Area of the Circle
The area \( A_c \) of the circle is given by:

\[ A_c = \pi r^2 \]

Calculating \( r^2 \):

\[ r^2 \approx (799.37)^2 \approx 638976.5 \text{ cm}^2 \]

Thus, the area of the circle is:

\[ A_c = \pi \times 638976.5 \approx 2001.6 \text{ cm}^2 \]

### Step 7: Calculate the Ratio of the Areas
Now, we can find the ratio of the area of the square to the area of the circle:

\[ \text{Ratio} = \frac{A_s}{A_c} = \frac{1576336}{2001.6} \]

Calculating this ratio gives us:

\[ \text{Ratio} \approx 786.5 \]

### Final Result
Thus, the ratio of the area of the square to that of the circle is approximately:

\[ \text{Ratio} \approx \frac{1576336}{2001.6} \approx 786.5 \]