Radians, arc length and sector area Cambridge IGCSE Additional Mathematics revision
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In plain words
Degrees are a human invention: 360 was chosen because it divides nicely. Radians are the natural measure: one radian is the angle at which the arc is exactly as long as the radius. Use radians, and the formulas for arcs and sectors become very short.
7 things to know
- π radians = 180°. To change degrees to radians, multiply by π/180. To change radians to degrees, multiply by 180/π.
- Arc length: s = rθ, with θ in radians.
- Area of a sector: A = (1/2)r²θ, with θ in radians.
- The perimeter of a sector is the arc plus two radii: rθ + 2r.
- Area of a triangle with two sides r and the angle θ between them: (1/2)r² sin θ.
- Area of a segment = area of the sector − area of the triangle = (1/2)r²(θ − sin θ).
- These formulas are not on the formula list, so they must be learned.
Worked example
A sector has radius 6 cm and angle 1.5 radians. Find its arc length, perimeter and area.
- Arc length = rθ = 6 × 1.5 = 9 cm.
- Perimeter = 9 + 6 + 6 = 21 cm.
- Area = (1/2) × 6² × 1.5 = 27 cm².
Worked example
Find the area of the segment cut off by a chord that subtends an angle of π/3 at the centre of a circle of radius 10 cm.
- Sector = (1/2) × 100 × π/3 = 52.36 cm².
- Triangle = (1/2) × 100 × sin(π/3) = 43.30 cm².
- Segment = 52.36 − 43.30 = 9.06 cm².
Tips and tricks
- Put the calculator into radian mode for this topic, and put it back afterwards.
- A perimeter always includes the straight edges. Forgetting the two radii is the most common error.
It lands in your notebook with its questions as flashcards.
Radians, arc length and sector area: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
How many radians is 180°?
A full turn is 2π radians.
What is the arc length of a sector with radius 4 and angle 2 radians?
s = rθ.
What is the area of a sector with radius 4 and angle 2 radians?
(1/2) × 16 × 2.
What is 60° in radians?
60 × π/180.
A sector has radius 5 and angle 1.2 radians. What is its perimeter?
The arc is 6, plus two radii of 5.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 8 marks. Write your answers on paper, then check them.
Radians, arc length and sector area
Cambridge IGCSE Additional Mathematics 0606 · 8 marks · papermunch.org
Name ______________________________ Date ______________
Convert 150° to radians, giving your answer in terms of π.[2]
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5π/6. 150 × π/180.
A sector has radius 8 cm and angle 0.75 radians. Find its arc length and its area.[3]
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Arc length 6 cm. Area 24 cm². 8 × 0.75, and (1/2) × 64 × 0.75.
A sector of a circle of radius 5 cm has an area of 40 cm². Find its angle and its arc length.[3]
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The angle is 3.2 radians and the arc length is 16 cm. (1/2) × 25 × θ = 40.
Answers: Radians, arc length and sector area
- 1. 5π/6. 150 × π/180.
- 2. Arc length 6 cm. Area 24 cm². 8 × 0.75, and (1/2) × 64 × 0.75.
- 3. The angle is 3.2 radians and the arc length is 16 cm. (1/2) × 25 × θ = 40.