The area of a minor sector subtending the central angle at the centre ` 40^@` is `8.25 cm^(2)`. What is the area of the remaining part (i.e., major sector ) of the circle ?

Options

A. `82.5 cm^(2) `
B. `74.25 cm^(2) `
C. `66 cm^(2) `
D. none of these

Answer

To find the area of the major sector of a circle when the area of the minor sector is given, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Central angle of the minor sector (θ) = 40 degrees
- Area of the minor sector = 8.25 cm²

2. **Use the Formula for the Area of a Sector**:
The area \( A \) of a sector of a circle can be calculated using the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.

3. **Set Up the Equation**:
We know the area of the minor sector is 8.25 cm², so we can set up the equation:
\[ \frac{40}{360} \times \pi r^2 = 8.25 \]

4. **Simplify the Equation**:
- Simplifying \( \frac{40}{360} \) gives \( \frac{1}{9} \):
\[ \frac{1}{9} \times \pi r^2 = 8.25 \]

5. **Solve for \( \pi r^2 \)**:
- Multiply both sides by 9:
\[ \pi r^2 = 8.25 \times 9 \]
- Calculate \( 8.25 \times 9 \):
\[ \pi r^2 = 74.25 \, \text{cm}^2 \]

6. **Calculate the Area of the Circle**:
The area of the entire circle is given by \( \pi r^2 \):
\[ \text{Area of Circle} = 74.25 \, \text{cm}^2 \]

7. **Find the Area of the Major Sector**:
The area of the major sector can be found by subtracting the area of the minor sector from the area of the circle:
\[ \text{Area of Major Sector} = \text{Area of Circle} - \text{Area of Minor Sector} \]
\[ \text{Area of Major Sector} = 74.25 \, \text{cm}^2 - 8.25 \, \text{cm}^2 \]
\[ \text{Area of Major Sector} = 66 \, \text{cm}^2 \]

### Final Answer:
The area of the remaining part (i.e., major sector) of the circle is **66 cm²**.

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