A circle field has perimeter 660m. Plot in the shape of a square having its vertices on the circumference is marked in the field. Calculate the area of the square field

Answer

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

### Step 1: Find the radius of the circle
The perimeter (circumference) of the circle is given as 660 m. The formula for the circumference of a circle is:
\[ C = 2\pi r \]
Setting this equal to the given perimeter:
\[ 2\pi r = 660 \]
To find the radius \( r \), we rearrange the equation:
\[ r = \frac{660}{2\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{660}{2 \times \frac{22}{7}} \]
Calculating this gives:
\[ r = \frac{660 \times 7}{2 \times 22} \]
\[ r = \frac{4620}{44} \]
\[ r = 105 \text{ m} \]

### Step 2: Find the diameter of the circle
The diameter \( d \) of the circle is twice the radius:
\[ d = 2r = 2 \times 105 = 210 \text{ m} \]

### Step 3: Relate the diameter to the square
The square inscribed in the circle has its diagonal equal to the diameter of the circle. Therefore, the diagonal \( d \) of the square is:
\[ d = 210 \text{ m} \]

### Step 4: Find the side length of the square
For a square, the relationship between the side length \( s \) and the diagonal \( d \) is given by:
\[ d = s\sqrt{2} \]
Rearranging to find \( s \):
\[ s = \frac{d}{\sqrt{2}} = \frac{210}{\sqrt{2}} \]
To simplify:
\[ s = 210 \times \frac{\sqrt{2}}{2} = 105\sqrt{2} \text{ m} \]

### Step 5: Calculate the area of the square
The area \( A \) of the square is given by:
\[ A = s^2 = (105\sqrt{2})^2 \]
Calculating this:
\[ A = 105^2 \times 2 = 11025 \times 2 = 22050 \text{ m}^2 \]

### Final Answer
The area of the square field is:
\[ \text{Area} = 22050 \text{ m}^2 \]

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