GCSE MATHS • HIGHER TIER
Arc Length Made Simple
Learn how to calculate the length of an arc using circumference, central angles and fractions of a full circle.
Clear method
Exam practice
Worked proofs
Arc Length in Four Moves
1
Start at North
Identify the radius or diameter and central angle.
2
Circumference
Calculate the circumference of the full circle.
3
Fraction
Write the central angle as a fraction of 360°.
4
Calculate
Multiply the circumference by the angle fraction.
What You’ll Learn
Understand Arc Length
Understand that an arc is part of a circle’s circumference.
Use Circle Vocabulary
Recognise radius, diameter, circumference, arc, sector and central angle.
Apply the Formula
Calculate arc length using the angle as a fraction of 360°.
Exact and Decimal Answers
Give answers in terms of π or as decimal approximations.
Key Rules
Arc length formula
Arc Length = (Angle ÷ 360) × Circumference.
Using the radius
Arc Length = (Angle ÷ 360) × 2πr
Full circle
A full circle contains 360°, so the angle determines the fraction of the circumference required.
Units
Arc length is a length, so use linear units such as cm, m or mm.
Worked Proof: Arc Length
Step-by-Step Method
- 1. Find the radius or diameter.
- 2. Calculate the circumference..
- 3. Find the fraction represented by the angle.
- 4. Multiply the circumference by this fraction. the angle.
- 5. Give the answer in exact or decimal form.
More Worked Examples
1
A circle has radius 10 cm and angle 180°.
Circumference = 20π cm
Arc Length = (180/360) × 20π
= 10π cm
= 10π cm
2
A circle has diameter 12 cm and angle 60°.
Circumference = 12π cm
Arc Length = (60/360) × 12π
= 2π cm
= 2π cm
3
A circle has radius 8 cm and angle 270°.
Circumference = 16π cm
Arc Length = (270/360) × 16π
= 12π cm
= 12π cm
4
A circle has radius 5 cm and angle 120°.
Circumference = 10π cm
Arc Length = (120/360) × 10π
= 10π/3 cm
= 10π/3 cm
- Common GCSE Mistakes
- Wrong measurement: Use circumference, not just the radius.
- Missing ÷ 360: Convert the angle into a fraction of 360°.
- Wrong angle: Use the central angle.
- Arc vs sector: Do not confuse arc length with sector area.
- Exam Tips
- Write the formula first: Use the arc length formula before substituting.
- Check radius or diameter: Use the correct circle measurement.
- Keep π exact: Leave π unless a decimal is requested.
Skills Used in Arc Length
Circle Area and Circumference
Circumference
Area of Circles
Area of a Sector
Geometry
Compound Shapes
Ready to Practise?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Arc Length Tutorial
Step-by-step walkthroughs of Arc Length with clear methods and exam tips.
Frequently Asked Questions
An arc is a section of the circumference of a circle.
Arc Length = (Angle ÷ 360) × Circumference.
A full circle contains 360°, so this gives the required fraction of the circumference.
Yes, unless a decimal answer is specifically requested.
Use linear units such as cm, m or mm.
The arc is the curved boundary of a sector and uses the same central angle.
