Area of major sector of angle `theta^(@)` of a circle with radius R will be `:`

Options

A. `( theta )/( 360) xx pi R^(2)`
B. `( theta )/( 180) xx pi R^(2)`
C. `((360-theta)/(360)) xx pi R^(2)`
D. None of these

Answer

To find the area of the major sector of a circle with radius \( R \) and central angle \( \theta \) degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Total Area of the Circle**:
The area \( A \) of a circle with radius \( R \) is given by the formula:
\[ A = \pi R^2 \]

**Hint**: Remember that the area of a circle is derived from the radius and the constant \( \pi \).

2. **Identify the Total Angle in a Circle**:
A complete circle has an angle of \( 360^\circ \). This means that the area of the circle corresponds to this full angle.

**Hint**: The total angle of a circle is always \( 360^\circ \).

3. **Calculate the Area of the Minor Sector**:
The area of the sector corresponding to the angle \( \theta \) can be calculated using the proportion of the angle \( \theta \) to the total angle \( 360^\circ \):
\[ \text{Area of minor sector} = \frac{\theta}{360} \times \pi R^2 \]

**Hint**: The area of a sector is a fraction of the total area based on the angle it subtends.

4. **Calculate the Area of the Major Sector**:
The area of the major sector is the remaining area of the circle after subtracting the area of the minor sector from the total area of the circle:
\[ \text{Area of major sector} = \text{Total area} - \text{Area of minor sector} \]
\[ \text{Area of major sector} = \pi R^2 - \frac{\theta}{360} \times \pi R^2 \]

5. **Simplify the Expression**:
Factor out \( \pi R^2 \):
\[ \text{Area of major sector} = \pi R^2 \left(1 - \frac{\theta}{360}\right) \]
\[ \text{Area of major sector} = \pi R^2 \left(\frac{360 - \theta}{360}\right) \]

**Hint**: When simplifying, look for common factors to factor out.

### Final Result:
The area of the major sector of angle \( \theta \) of a circle with radius \( R \) is:
\[ \text{Area of major sector} = \frac{(360 - \theta)}{360} \times \pi R^2 \]