Arc length formula: l = rθ
The arc length formula is arc length l = rθ, with θ in radians.
What each letter means
- \(l\) the length of the arc
- \(r\) the radius
- \(\theta\) the angle at the centre, in radians
When to use it
To find the curved edge of a sector or part of a circle, for example the perimeter of a sector (arc length + 2r).
Worked example
A sector of a circle of radius 6 cm has perimeter 20 cm. Find the angle of the sector in radians.
- Arc length \(=20-2\times6=8\)
- \(8=6\theta\)
Answer: \(\theta=\tfrac43\) radians
Common mistake
Putting θ in degrees into l = rθ. Convert first: θ in radians = θ° × π/180.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | Not in the Edexcel booklet: learn it |
| Edexcel International A Level | Not in the booklet: learn it |
| AQA or OCR A Level | Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page |
From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).
Practise and revise
- Practise A Level Maths practice
- Revise Trigonometry
- Revise IAL P1 trigonometry
- Print Edexcel A Level (9MA0) one-page formula sheet
Questions
What is the arc length formula?
The arc length formula is arc length l = rθ, with θ in radians. l: the length of the arc; r: the radius; θ: the angle at the centre, in radians.
Is the arc length given in the exam?
Edexcel A Level (9MA0): not in the Edexcel booklet: learn it. Edexcel International A Level: not in the booklet: learn it. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.
How do I find the perimeter of a sector?
Add the arc length to the two straight edges: perimeter = arc length + 2r.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.