Ch 1 · 9 Circular Measure · O Level Additional Mathematics 4037
Topic 1 of 2 in Trigonometry — O Level Additional Mathematics 4037 — 18 lessons.
Introduction
Fan blades, slices of cake, a pendulum's swing: each sweeps out a sector of a circle. In Additional Mathematics angles in these problems are measured in radians, which makes the formulas for arc length and area very simple. You must remember the formulas, since they are not given, and many questions combine sectors with triangles into compound shapes. (Syllabus topic 9.)
Key ideas
One radian is the angle at the centre made by an arc equal in length to the radius. A full turn is 2π radians:
🧑🏫 Teacher's explanation
Imagine you are walking along the edge of a circular running track. If you walk a distance exactly equal to the radius of the track, the angle you have swept out from the center is exactly 1 radian. It is a 'natural' way to measure angles because it links the angle directly to the distance traveled along the curve, unlike degrees which are arbitrary divisions of a circle.
📖 Definitions
🔍 Worked example — Easy
Convert 150^° into radians, leaving your answer in terms of π.
🔍 Worked example — Standard
A sector of a circle has a radius of 8 cm and an angle of 3π/4 radians. Calculate the perimeter of the sector.
🔍 Worked example — Exam-style
A chord of a circle with radius 10 cm subtends an angle of 2π/3 at the centre. Find the area of the segment formed by this chord.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
A sector has radius 6 cm and angle 2π/3. Arc = 6 × 2π/3 = 4π cm; area = tfrac12 × 36 × 2π/3 = 12π cm²; perimeter = 12 + 4π cm.
🔍 Worked example — Easy
A sector of a circle has a radius of 6 cm and an angle of 2π/3 radians. Find the exact arc length, the area of the sector, and the area of the segment cut off by the chord.
🔍 Worked example — Standard
A sector of a circle has a radius of 5 cm and a perimeter of 17 cm. Find the angle θ in radians and the area of the sector.
🔍 Worked example — Exam-style
A shape is formed by taking a square OABC of side length 4 cm and removing a quarter-circle of radius 4 cm centered at O. Find the area and the perimeter of the remaining shape.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 5π/4; 75^°. (2) tfrac12 × 81θ = 27π ⇒ θ = 2π/3; arc = 6π cm. (3) tfrac12 × 100 (π/3 - sqrt3/2 ) = 50π/3 - 25sqrt3 cm². (4) 2r + 0.8r = 28 ⇒ r = 10; area = tfrac12 × 100 × 0.8 = 40 cm².