From a circular piece of cardboard of radius 3 cm, two sectors of `40^@` each have been cut off. The area of the remaining portion is : `(use pi = 22/7)`

Options

A. 11 `cm^2`
B. 22 `cm^2`
C. 33 `cm^2`
D. 44 `cm^2`

Answer

To find the area of the remaining portion of the circular piece of cardboard after cutting off two sectors of 40 degrees each, we can follow these steps:

### Step 1: Calculate the total area of the circle
The formula for the area of a circle is:
\[ \text{Area} = \pi r^2 \]
Given that the radius \( r = 3 \) cm and using \( \pi = \frac{22}{7} \):
\[ \text{Area} = \frac{22}{7} \times (3)^2 \]
\[ = \frac{22}{7} \times 9 \]
\[ = \frac{198}{7} \text{ cm}^2 \]

### Step 2: Calculate the total angle of the sectors cut off
Each sector has an angle of \( 40^\circ \), and since two sectors are cut off:
\[ \text{Total angle of sectors} = 40^\circ + 40^\circ = 80^\circ \]

### Step 3: Calculate the area of the sectors cut off
Using the formula for the area of a sector:
\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]
For the total area of the two sectors:
\[ \text{Area of sectors} = \frac{80}{360} \times \frac{22}{7} \times (3)^2 \]
\[ = \frac{80}{360} \times \frac{22}{7} \times 9 \]
\[ = \frac{2}{9} \times \frac{22}{7} \times 9 \]
\[ = \frac{2 \times 22}{7} = \frac{44}{7} \text{ cm}^2 \]

### Step 4: Calculate the area of the remaining portion
Now, subtract the area of the sectors from the total area of the circle:
\[ \text{Area of remaining portion} = \text{Total area} - \text{Area of sectors} \]
\[ = \frac{198}{7} - \frac{44}{7} \]
\[ = \frac{198 - 44}{7} = \frac{154}{7} \text{ cm}^2 \]

### Step 5: Simplify the area of the remaining portion
Calculating \( \frac{154}{7} \):
\[ = 22 \text{ cm}^2 \]

### Final Answer
The area of the remaining portion is \( 22 \text{ cm}^2 \).

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