This article covers the essential skills of arc length, sector area, and radian measure – a core topic in DSE Paper 1 Section A(2) and Section B worth 4–6 marks. You will learn the definition of radian measure, how to convert between degrees and radians, the formulas for arc length and sector area in both degree and radian measure, how to calculate segment area, and how to solve composite figure problems. The article includes step-by-step worked examples, DSE exam techniques, practice questions. These are essential skills that appear regularly in DSE papers.
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Mensuration | Arc Length, Sector Area & Radian Measure
Question 1 MC
Convert \( 150^\circ \) to radians.
A. \( \frac{5\pi}{6} \) B. \( \frac{6\pi}{5} \) C. \( \frac{2\pi}{3} \) D. \( \frac{\pi}{6} \)
Question 2 MC
A circle has radius 10 cm. Find the arc length for a central angle of \( 72^\circ \).
A. \( 2\pi \) cm B. \( 4\pi \) cm C. \( 8\pi \) cm D. \( 12\pi \) cm
Question 3 Short Answer
Find the area of a sector with radius 6 cm and central angle \( \frac{\pi}{6} \) radians.
Question 4 Short Answer
A segment has radius 8 cm and central angle \( \frac{\pi}{3} \) radians. Find its area.
Question 5 MC
A sector has area \( 18\pi \) cm² and radius 6 cm. Find the central angle in radians.
A. \( \frac{\pi}{2} \) B. \( \pi \) C. \( 2\pi \) D. \( \frac{\pi}{4} \)
Question 6 Short Answer
A circle has radius 8 cm. Find the perimeter of a sector with central angle \( \frac{\pi}{3} \) radians.
Question 7 MC
Convert \( \frac{3\pi}{4} \) radians to degrees.
A. \( 135^\circ \) B. \( 120^\circ \) C. \( 150^\circ \) D. \( 90^\circ \)
Question 8 Short Answer
A figure consists of a sector of radius 10 cm and angle \( 45^\circ \) attached to a rectangle of width 10 cm and height 4 cm. Find the total area.
Question 9 Short Answer
Find the area of a segment cut off by a chord in a circle of radius 3 cm with central angle \( 120^\circ \). (Use radians)
Question 10 Short Answer
A sector has perimeter 20 cm and radius 6 cm. Find the central angle in radians.
When using the arc length formula \( s = r\theta \), ensure \( \theta \) is in radians, not degrees. For degree measure, use \( s = \frac{\theta}{360^\circ} \times 2\pi r \).