The area of the greatest circle, which can be inscribed in a square whose perimeter is 120 cm, is :

Options

A. `22/7 xx 15^2 cm^2`
B. `22/7 xx (7/2)^2 cm^2`
C. `22/7 xx (15/2)^2 cm^2`
D. `22/7 xx (9/2)^2 cm^2`

Answer

To find the area of the greatest circle that can be inscribed in a square with a perimeter of 120 cm, we can follow these steps:

### Step 1: Find the side length of the square
The perimeter \( P \) of a square is given by the formula:
\[ P = 4 \times \text{side length} \]
Given that the perimeter is 120 cm, we can set up the equation:
\[ 4a = 120 \]
where \( a \) is the side length of the square.

### Step 2: Solve for the side length
To find \( a \), we divide both sides of the equation by 4:
\[ a = \frac{120}{4} = 30 \text{ cm} \]
So, the side length of the square is 30 cm.

### Step 3: Determine the radius of the inscribed circle
The diameter of the inscribed circle is equal to the side length of the square. Therefore, the radius \( r \) can be calculated as:
\[ r = \frac{\text{diameter}}{2} = \frac{a}{2} = \frac{30}{2} = 15 \text{ cm} \]

### Step 4: Calculate the area of the circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Using \( \pi \approx \frac{22}{7} \) for calculation, we substitute the radius:
\[ A = \frac{22}{7} \times (15)^2 \]
Calculating \( (15)^2 \):
\[ (15)^2 = 225 \]
Now substituting back into the area formula:
\[ A = \frac{22}{7} \times 225 \]

### Step 5: Simplify the area calculation
To simplify:
\[ A = \frac{22 \times 225}{7} \]
Calculating \( 22 \times 225 \):
\[ 22 \times 225 = 4950 \]
Now substituting:
\[ A = \frac{4950}{7} \text{ cm}^2 \]

### Final Answer
The area of the greatest circle that can be inscribed in the square is:
\[ A \approx 707.14 \text{ cm}^2 \]