The perimeter of square and a circular field are the same. If the area of the circular field is 3850 `m^2`, then what is the area (in `m^2`) of the square?

Options

A. 4225
B. 3025
C. 2500
D. 2025

Answer

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

### Step 1: Understand the relationship between the perimeter of the square and the circular field.
The problem states that the perimeter of the square is equal to the circumference of the circular field.

### Step 2: Write the formulas for the perimeter and circumference.
- The perimeter of a square (P_square) is given by:
\[ P_{\text{square}} = 4 \times \text{side} \]
- The circumference of a circle (C_circle) is given by:
\[ C_{\text{circle}} = 2 \pi r \]

### Step 3: Set the perimeters equal to each other.
Since the perimeter of the square is equal to the circumference of the circular field, we can write:
\[ 4 \times \text{side} = 2 \pi r \]

### Step 4: Find the radius of the circular field using its area.
We know the area (A) of the circular field is given as:
\[ A = \pi r^2 = 3850 \, m^2 \]
To find the radius \( r \), we rearrange the formula:
\[ r^2 = \frac{A}{\pi} \]
Substituting the area:
\[ r^2 = \frac{3850}{\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r^2 = \frac{3850 \times 7}{22} = \frac{26950}{22} = 1225 \]
Now, taking the square root:
\[ r = \sqrt{1225} = 35 \, m \]

### Step 5: Substitute the radius back into the perimeter equation.
Now that we have the radius, we can substitute \( r \) back into the perimeter equation:
\[ 4 \times \text{side} = 2 \pi r \]
Substituting \( r = 35 \):
\[ 4 \times \text{side} = 2 \times \frac{22}{7} \times 35 \]
Calculating the right side:
\[ = 2 \times \frac{22 \times 35}{7} = 2 \times 110 = 220 \]
So we have:
\[ 4 \times \text{side} = 220 \]
Dividing both sides by 4:
\[ \text{side} = \frac{220}{4} = 55 \, m \]

### Step 6: Calculate the area of the square.
The area \( A_{\text{square}} \) of the square is given by:
\[ A_{\text{square}} = \text{side}^2 = 55^2 = 3025 \, m^2 \]

### Final Answer:
The area of the square is \( 3025 \, m^2 \).

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