Question

Length of an arc of a sector of a circle with radius \(r\) and sectorial angle \(\theta\) (measured in degrees) is

• A. \(\dfrac{\pi r\theta}{360}\)
• B. \(\dfrac{\pi r\theta}{180}\)
• C. \(\dfrac{\pi r^2\theta}{360}\)
• D. \(\dfrac{\pi r^2\theta}{180}\)

Hint:

For arc length with angle in degrees, always use \[ \text{Arc length} = \frac{\theta}{360}\times 2\pi r = \frac{\pi r\theta}{180} \] Do not confuse it with the area of a sector, which uses \(r^2\).

Answer 1

The Correct Option is B

Step 1: Recall the formula for the circumference of a circle.}
The total circumference of a circle of radius \(r\) is: \[ 2\pi r \] This corresponds to a central angle of \(360^\circ\).

Step 2: Find the arc length for angle \(\theta\).}
If the full circle of \(360^\circ\) has arc length \(2\pi r\), then a sector with angle \(\theta\) will have arc length: \[ \frac{\theta}{360} \times 2\pi r \]
Step 3: Simplify the formula.}
\[ \text{Arc length} = \frac{\theta}{360} \times 2\pi r \] \[ = \frac{2\pi r\theta}{360} \] \[ = \frac{\pi r\theta}{180} \] So, the required length of the arc is: \[ \frac{\pi r\theta}{180} \]
Step 4: Compare with the given options.}

• (A) \(\dfrac{\pi r\theta}{360}\): Incorrect. This is half the required value.
• (B) \(\dfrac{\pi r\theta}{180}\): Correct.
• (C) \(\dfrac{\pi r^2\theta}{360}\): Incorrect. This resembles an area-related expression, not arc length.
• (D) \(\dfrac{\pi r^2\theta}{180}\): Incorrect. Arc length depends on \(r\), not \(r^2\).

Therefore, the correct option is \((B)\).
Final Answer:} \(\dfrac{\pi r\theta}{180}\).