From a rectangular metal sheet of length 11 cm and breadth 8 cm , three circular plates of radii 3 cm , 2 cm , and 1 cm are cut . The area of the remaining part is `( "in" "cm^(2))` equal to the area of a circle , then the area of the circle is ( in cm) ______.

Answer

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

### Step 1: Calculate the area of the rectangular metal sheet.
The area \( A \) of a rectangle is given by the formula:
\[ A = \text{length} \times \text{breadth} \]
Given:
- Length = 11 cm
- Breadth = 8 cm

Calculating the area:
\[ A = 11 \, \text{cm} \times 8 \, \text{cm} = 88 \, \text{cm}^2 \]

### Step 2: Calculate the area of the three circular plates.
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Where \( r \) is the radius of the circle.

#### Area of the first circle (radius = 3 cm):
\[ A_1 = \pi (3)^2 = 9\pi \, \text{cm}^2 \]

#### Area of the second circle (radius = 2 cm):
\[ A_2 = \pi (2)^2 = 4\pi \, \text{cm}^2 \]

#### Area of the third circle (radius = 1 cm):
\[ A_3 = \pi (1)^2 = 1\pi \, \text{cm}^2 \]

### Step 3: Calculate the total area of the three circular plates.
\[ \text{Total Area of circles} = A_1 + A_2 + A_3 = 9\pi + 4\pi + 1\pi = 14\pi \, \text{cm}^2 \]

### Step 4: Calculate the area of the remaining part of the metal sheet.
\[ \text{Area of remaining part} = \text{Area of rectangle} - \text{Total Area of circles} \]
\[ = 88 \, \text{cm}^2 - 14\pi \, \text{cm}^2 \]

### Step 5: Set the area of the remaining part equal to the area of a circle.
Let the radius of the circle whose area is equal to the remaining part be \( r \). Then:
\[ \pi r^2 = 88 - 14\pi \]

### Step 6: Solve for \( r^2 \).
Dividing both sides by \( \pi \):
\[ r^2 = \frac{88 - 14\pi}{\pi} \]

### Step 7: Calculate the area of the circle.
The area of the circle is given by:
\[ \text{Area} = \pi r^2 \]
Substituting \( r^2 \):
\[ \text{Area} = \pi \left(\frac{88 - 14\pi}{\pi}\right) = 88 - 14\pi \]

### Step 8: Substitute \( \pi \) with \( \frac{22}{7} \) to find the numerical value.
\[ \text{Area} = 88 - 14 \times \frac{22}{7} \]
Calculating \( 14 \times \frac{22}{7} \):
\[ = \frac{308}{7} \]
Now, substituting back:
\[ \text{Area} = 88 - \frac{308}{7} = \frac{616}{7} - \frac{308}{7} = \frac{308}{7} \]
Calculating \( \frac{308}{7} \):
\[ = 44 \, \text{cm}^2 \]

Thus, the area of the circle is \( 44 \, \text{cm}^2 \).