A taxi firm paints its logo as a coloured sector of a circle on each rear car door. The sector has radius \(9\) cm and angle \(120^{\circ}\), measured from ...
Question 1 Report
A taxi firm paints its logo as a coloured sector of a circle on each rear car door. The sector has radius \(9\) cm and angle \(120^{\circ}\), measured from the centre of the circle.
- Work out the arc length of the sector. Give your answer as a multiple of \(\pi\). (2)
Answer Details
The arc length of a sector is the fraction of the full circle's circumference that the sector's angle represents: \( \text{arc length} = \dfrac{\theta}{360} \times 2\pi r \), where \( \theta \) is the sector angle at the centre.
(a) Substituting \( \theta = 120^\circ \) and \( r = 9 \) cm: \( \text{arc length} = \dfrac{120}{360} \times 2\pi \times 9 \) [1 mark]. Since \( \dfrac{120}{360} = \dfrac{1}{3} \), this simplifies to \( \dfrac{1}{3} \times 18\pi = 6\pi \) cm, left as a multiple of \( \pi \) as requested [1 mark].
The fraction \( \dfrac{\theta}{360} \) always compares the sector's angle with a full turn; forgetting to convert the angle into this fraction (and instead using \( \theta \) directly) is the usual mistake with sector questions.
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