The central angle of a sector is `240^@` and radius is 12 cm, then the area (in sq cm ) of the sector is

Options

A. ` 26 pi `
B. ` 24 pi`
C. ` 20 pi`
D. ` 18 pi`

Answer

To find the area of a sector with a central angle of \(240^\circ\) and a radius of \(12 \, \text{cm}\), we can use the formula for the area of a sector:

\[ \text{Area of sector} = \frac{\theta}{360} \times \pi R^2 \]

where:
- \(\theta\) is the central angle in degrees,
- \(R\) is the radius of the circle.

### Step-by-Step Solution:

1. **Identify the values**:
- Central angle \(\theta = 240^\circ\)
- Radius \(R = 12 \, \text{cm}\)

2. **Substitute the values into the formula**:
\[ \text{Area of sector} = \frac{240}{360} \times \pi \times (12)^2 \]

3. **Calculate \(R^2\)**:
\[ (12)^2 = 144 \]

4. **Substitute \(R^2\) back into the formula**:
\[ \text{Area of sector} = \frac{240}{360} \times \pi \times 144 \]

5. **Simplify \(\frac{240}{360}\)**:
\[ \frac{240}{360} = \frac{2}{3} \]

6. **Now substitute this back into the area formula**:
\[ \text{Area of sector} = \frac{2}{3} \times \pi \times 144 \]

7. **Calculate \(\frac{2}{3} \times 144\)**:
\[ \frac{2 \times 144}{3} = \frac{288}{3} = 96 \]

8. **Final area of the sector**:
\[ \text{Area of sector} = 96\pi \, \text{cm}^2 \]

Thus, the area of the sector is \(96\pi \, \text{cm}^2\).