The area (in `cm^2`) of a sector of a circle of radius 2 cm is `(4pi)/5`. What is the central angle (in degrees) of the sector?

Options

A. 60
B. 75
C. 36
D. 72

Answer

To find the central angle of a sector of a circle given the area of the sector and the radius, we can use the formula for the area of a sector:

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

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

### Step-by-step Solution:

1. **Identify the given values**:
- Radius \(r = 2 \, \text{cm}\)
- Area of the sector = \(\frac{4\pi}{5} \, \text{cm}^2\)

2. **Substitute the values into the area formula**:
\[ \frac{4\pi}{5} = \frac{\theta}{360^\circ} \times \pi (2^2) \]

3. **Calculate \(r^2\)**:
\[ r^2 = 2^2 = 4 \]
Therefore, the equation becomes:
\[ \frac{4\pi}{5} = \frac{\theta}{360^\circ} \times \pi \times 4 \]

4. **Simplify the equation**:
\[ \frac{4\pi}{5} = \frac{\theta \cdot 4\pi}{360^\circ} \]

5. **Cancel \(\pi\) from both sides**:
\[ \frac{4}{5} = \frac{\theta \cdot 4}{360^\circ} \]

6. **Multiply both sides by \(360^\circ\)**:
\[ \frac{4 \cdot 360^\circ}{5} = 4\theta \]

7. **Divide both sides by 4**:
\[ \frac{360^\circ}{5} = \theta \]

8. **Calculate \(\theta\)**:
\[ \theta = 72^\circ \]

### Final Answer:
The central angle of the sector is \(72^\circ\).