Arc length is a key topic in GCSE Maths that appears on both Foundation and Higher papers across AQA, Edexcel, and OCR. An arc is a portion of the circumference of a circle, and finding its length requires combining the circumference formula with the fraction of the circle represented by the angle. This guide walks you through the formula, three worked examples, and common pitfalls.
What Is Arc Length?
An arc is a curved section of the circumference of a circle. The arc length depends on two things: the radius of the circle and the angle at the centre that the arc subtends. A larger angle gives a longer arc; a full 360° angle gives the entire circumference.
Key Formulas
Here θ is the angle at the centre in degrees and r is the radius.
Step-by-Step Method
- Identify the radius (r) and the angle at the centre (θ) from the question or diagram.
- Write the formula: Arc length = (θ / 360) × 2πr.
- Substitute the values of θ and r into the formula.
- Calculate the answer, giving it to the degree of accuracy requested (usually 1 decimal place or in terms of π).
Worked Example 1 — Foundation Level
Question: A sector has a radius of 8 cm and an angle of 90°. Find the arc length. Give your answer to 1 decimal place.
Working:
Arc length = (90 / 360) × 2 × π × 8
= (1/4) × 16π
= 4π
= 12.566...
Answer: Arc length = 12.6 cm (1 d.p.)
Worked Example 2 — Higher Level
Question: An arc of a circle with radius 12 cm has a length of 15 cm. Find the angle at the centre. Give your answer to 1 decimal place.
Working:
Arc length = (θ / 360) × 2πr
15 = (θ / 360) × 2 × π × 12
15 = (θ / 360) × 24π
θ / 360 = 15 / (24π)
θ = 360 × 15 / (24π) = 5400 / (24π) = 71.619...
Answer: θ = 71.6° (1 d.p.)
Worked Example 3 — Exam Style
Question: A sector with radius 10 cm has an arc length of 8π cm. Find the angle at the centre. Give your answer as an exact value.
Working:
Arc length = (θ / 360) × 2πr
8π = (θ / 360) × 2 × π × 10
8π = (θ / 360) × 20π
Divide both sides by π: 8 = (θ / 360) × 20
θ / 360 = 8 / 20 = 2/5
θ = 360 × 2/5 = 144
Answer: θ = 144°
Common Mistakes
- Using diameter instead of radius. The formula uses radius. If the question gives the diameter, halve it first before substituting.
- Forgetting to multiply by 2. The circumference formula is 2πr, not πr. Missing the 2 halves your answer.
- Rounding too early. Keep full calculator values until the final step, then round to the required accuracy.
Exam Tips
- If the answer is requested "in terms of π", leave π in your final answer and simplify the numerical part.
- Always check whether the question gives a radius or a diameter — this is one of the most common sources of lost marks.
- Arc length questions can be combined with sector area questions. Make sure you use the correct formula for each part.
- Show your substitution step clearly — this earns a method mark even if your final answer is wrong.
Practice Questions
Q1 (Foundation): Find the arc length of a sector with radius 5 cm and angle 72°. Give your answer to 1 d.p.
Q2 (Foundation): A sector has a radius of 14 cm and an angle of 180°. Find the arc length in terms of π.
Q3 (Higher): A sector has an arc length of 20 cm and a radius of 9 cm. Find the angle at the centre to 1 d.p.
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Summary
- Arc length is the curved distance along part of a circle's circumference. The formula is (θ / 360) × 2πr. You need to know the radius and the angle at the centre. You can also rearrange the formula to find the angle or the radius when the arc length is given. Always check whether the question provides a radius or diameter, and show your substitution step for method marks.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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