Convert angles to radians and find arc lengths
Convert between degrees and radians, then use the radian definition to calculate arc length in exact form.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the radian measure of 90°?
What you will learn
- Convert between degrees and radians
- Use s = rθ when θ is measured in radians
- Find a missing radius or angle from an arc length
A radian measures angle through arc length
One radian is the central angle that subtends an arc equal in length to the radius. A full turn has circumference 2πr and angle 2π radians, so π radians = 180°.
Convert degrees to radians by multiplying by π/180; convert radians to degrees by multiplying by 180/π. Keep exact multiples of π until the question asks for a decimal.
Arc length uses radians
For radius r and angle θ in radians, s = rθ. If the angle is given in degrees, convert it first; substituting degree values directly gives the wrong length.
Rearrange as θ = s/r or r = s/θ when a different quantity is unknown. The angle is dimensionless, so s has the same length unit as r.
For a full circumference θ = 2π, so s = 2πr. For a semicircle θ = π, so s = πr.
Pause and explain
Which angle unit must be used in s = rθ?
Worked example
Find the arc length for radius 12 cm and central angle 150°.
Show the worked solution
- Convert: 150° × π/180 = 5π/6 radians.
- Use s = rθ with r = 12.
- s = 12 × 5π/6 = 10π cm.
Answer 10π cm.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the arc length for r = 7 cm and θ = 1.2 radians.
Give me a hint
The angle is already in radians, so multiply it by the radius.
Compare my reasoning
- Use s = rθ.
- s = 7 × 1.2 = 8.4.
- The arc length is 8.4 cm.
8.4 cm.
Look for these in your work
- Radians recognised
- Formula applied
- Length unit retained
Find the arc length of a radius-6 m circle subtended by 210°.
Give me a hint
Convert 210° to radians before multiplying by 6.
Compare my reasoning
- 210° × π/180 = 7π/6 radians.
- s = rθ = 6 × 7π/6.
- Simplify the exact length.
7π m.
Look for these in your work
- Angle converted
- Arc formula used
- Exact answer simplified
A wheel has radius 0.35 m and turns through 5 radians. How far does a point on its rim travel?
Give me a hint
The rim distance is the arc length rθ.
Compare my reasoning
- The angle is already in radians.
- s = 0.35 × 5 = 1.75.
- The point travels 1.75 m along the rim.
1.75 m.
Look for these in your work
- Radius used
- Angle unit checked
- Distance interpreted
Common mistakes
- Substituting degrees into s = rθ
- Dropping π during conversion
- Giving a length in square units
- Rounding before an exact answer is simplified
What can you explain now?
How many radians are in 180°?
Compare with the explanation
π radians
A half-turn is π radians; a full turn is 2π radians.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why must θ be in radians when using s = rθ?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.