The area of the greatest circle inscribed inside a square of side 21 cm is (Take `pi = (22)/(7)`)

Options

A. `344.5 cm^(2)`
B. `364.5 cm^(2)`
C. `346.5 cm^(2)`
D. `366.5 cm^(2)`

Answer

To find the area of the greatest circle inscribed inside a square of side 21 cm, we can follow these steps:

### Step 1: Determine the radius of the circle
The radius of the greatest circle that can be inscribed in a square is half the length of the side of the square.

Given:
- Side of the square = 21 cm

Radius (r) = Side of the square / 2
\[ r = \frac{21 \text{ cm}}{2} = 10.5 \text{ cm} \]

### Step 2: Use the formula for the area of a circle
The area (A) of a circle is given by the formula:
\[ A = \pi r^2 \]

### Step 3: Substitute the values into the area formula
We know:
- \( \pi = \frac{22}{7} \)
- \( r = 10.5 \text{ cm} \)

Now, substituting these values into the area formula:
\[ A = \frac{22}{7} \times (10.5)^2 \]

### Step 4: Calculate \( (10.5)^2 \)
First, calculate \( (10.5)^2 \):
\[ (10.5)^2 = 110.25 \]

### Step 5: Substitute \( (10.5)^2 \) back into the area formula
Now substitute \( 110.25 \) back into the area formula:
\[ A = \frac{22}{7} \times 110.25 \]

### Step 6: Perform the multiplication
Calculate:
\[ A = \frac{22 \times 110.25}{7} \]

### Step 7: Calculate \( 22 \times 110.25 \)
\[ 22 \times 110.25 = 2425.5 \]

### Step 8: Divide by 7
Now divide \( 2425.5 \) by \( 7 \):
\[ A = \frac{2425.5}{7} = 346.5 \text{ cm}^2 \]

### Final Answer
The area of the greatest circle inscribed inside the square is:
\[ \text{Area} = 346.5 \text{ cm}^2 \]

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