A sector has an arc length of `4pi` and a radius of 3. What is the central angle of the sector?

Answer

To find the central angle of a sector given the arc length and radius, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- Arc length (L) = \(4\pi\)
- Radius (R) = 3

2. **Use the formula for arc length**:
The formula for the arc length of a sector is given by:
\[ L = R \times \theta \]
where \(L\) is the arc length, \(R\) is the radius, and \(\theta\) is the central angle in radians.

3. **Substitute the known values into the formula**:
\[ 4\pi = 3 \times \theta \]

4. **Solve for \(\theta\)**:
To isolate \(\theta\), divide both sides by 3:
\[ \theta = \frac{4\pi}{3} \]

5. **Convert \(\theta\) from radians to degrees**:
To convert radians to degrees, use the conversion factor:
\[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]
Therefore:
\[ \theta = \frac{4\pi}{3} \times \frac{180}{\pi} \]

6. **Simplify the expression**:
The \(\pi\) in the numerator and denominator cancels out:
\[ \theta = \frac{4 \times 180}{3} \]

7. **Calculate the value**:
First, simplify \( \frac{180}{3} = 60\):
\[ \theta = 4 \times 60 = 240 \]

8. **Final answer**:
The central angle \(\theta\) is \(240^\circ\).