What is the angle subtended at the centre of a circle of radius 5cm by an arc length `4pi` cm?

Answer

To find the angle subtended at the center of a circle by an arc of length \(4\pi\) cm, we can use the formula for the length of an arc:

\[ L = \frac{\theta}{360} \times 2\pi r \]

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

### Step-by-Step Solution:

1. **Identify the given values**:
- Length of the arc \(L = 4\pi\) cm
- Radius of the circle \(r = 5\) cm

2. **Substitute the values into the arc length formula**:
\[ 4\pi = \frac{\theta}{360} \times 2\pi \times 5 \]

3. **Simplify the equation**:
- First, calculate \(2\pi \times 5\):
\[ 2\pi \times 5 = 10\pi \]
- Now substitute this back into the equation:
\[ 4\pi = \frac{\theta}{360} \times 10\pi \]

4. **Cancel \(\pi\) from both sides**:
\[ 4 = \frac{\theta}{360} \times 10 \]

5. **Multiply both sides by 360 to eliminate the fraction**:
\[ 4 \times 360 = 10\theta \]
\[ 1440 = 10\theta \]

6. **Divide both sides by 10 to solve for \(\theta\)**:
\[ \theta = \frac{1440}{10} = 144 \]

7. **Conclusion**:
The angle subtended at the center of the circle by the arc of length \(4\pi\) cm is \(144^\circ\).