In a circle of radius r cm, an arc subtends an angle of `60^(@)` at the centre. The length of the arc will be `:`

Options

A. `( pi r)/( 3) cm`
B. `( 2 pi r )/( 3) cm^(2) `
C. `( pi r^(2))/( 3) cm^(2)`
D. `( pi r )/( 3) cm^(2)`

Answer

To find the length of an arc in a circle, we can use the formula:

\[ \text{Length of arc} = r \times \theta \]

where:
- \( r \) is the radius of the circle,
- \( \theta \) is the angle subtended by the arc at the center of the circle (in radians).

### Step-by-Step Solution:

1. **Identify the given values**:
- Radius of the circle: \( r \) cm
- Angle subtended at the center: \( 60^\circ \)

2. **Convert the angle from degrees to radians**:
- To convert degrees to radians, we use the conversion factor:
\[ \text{radians} = \frac{\text{degrees} \times \pi}{180} \]
- For \( 60^\circ \):
\[ \theta = \frac{60 \times \pi}{180} = \frac{\pi}{3} \text{ radians} \]

3. **Substitute the values into the arc length formula**:
- Now that we have \( r \) and \( \theta \):
\[ \text{Length of arc} = r \times \frac{\pi}{3} \]

4. **Final expression for the length of the arc**:
- Thus, the length of the arc \( AB \) is:
\[ \text{Length of arc} = \frac{\pi r}{3} \text{ cm} \]

### Final Answer:
The length of the arc is \( \frac{\pi r}{3} \) cm.

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