GCSE MATHS • HIGHER TIER
Angles at the Centre Made Simple
Learn how to use the relationship between angles at the centre and angles at the circumference to solve GCSE circle theorem questions.
Clear method
Exam practice
Worked proofs
Angles at the Centre in Four Moves
1
Identify
Identify the centre angle and circumference angle.
2
Check
Check that both angles stand on the same arc.
3
Apply
Use the doubling relationship between the two angles.
4
Solve
Calculate the missing angle or solve the algebra.
What You’ll Learn
Understand the Theorem
Understand how angles at the centre and circumference are related.
Recognise the Same Arc
Check that both angles stand on the same arc.
Apply the Key Rule
Double or halve angles correctly.
Solve Algebraic Questions
Use the theorem to form and solve equations.
Connect Circle Theorems
Use this rule alongside other GCSE circle theorems.
Key Rules
Angle at the Centre = 2 × Angle at the Circumference
If you know one angle, you can always find the other using this relationship.
Understanding the Theorem : The centre of the circle is the point exactly in the middle. The circumference is the outer edge. When two angles are formed from the same arc, the angle at the centre is always double the angle at the circumference.
Worked Proof: Angles at the Centre
Step-by-Step Method
- 1. Identify the centre of the circle.
- 2. Identify the angle at the circumference.
- 3. Check that both angles stand on the same arc.
- 4. Apply the theorem.
- 5. Solve any calculations or algebra.
- 6. Check the answer is reasonable.
More Worked Examples
1
What is the angle at the centre theorem?
Angle at the centre = 2 × 48
= 96°
2
Question: The angle at the centre is 120°.
Angle at the circumference = 120 ÷ 2
= 60°
3
Question: The angle at the centre is 150°.
Angle at the circumference = 150 ÷ 2
= 75°
4
Question: The angle at the circumference is x and the angle at the centre is 100°.
2x = 100
x = 50
- Common GCSE Mistakes
- Doubling rule: Remember the centre angle is double.
- Wrong arc: Check both angles stand on the same arc.
- Missing centre: Identify the centre point correctly.
- Algebra errors: Solve the angle equation carefully.
- Wrong operation: Multiply or divide in the correct direction.
- Exam Tips
- Check the arc: Confirm both angles use the same arc.
- State the theorem: Name the rule in reasoning questions.
- Centre to circumference: Divide by 2.
- Circumference to centre: Multiply by 2.
- Show algebra clearly: Write the equation first.
Skills Used in Angles at the Centre
Angles in the Same Segment
Cyclic Quadrilaterals
Tangents
Alternate Segment Theorem
Radius and Tangent Theorem
Circle Geometry
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 Angles at the Centre Tutorial
Step-by-step walkthroughs of Angles at the Centre with clear methods and exam tips.
Frequently Asked Questions
The angle at the centre is twice the angle at the circumference when both stand on the same arc.
Multiply the circumference angle by 2.
Divide the centre angle by 2.
The doubling relationship applies when both angles stand on the same arc.
Yes. Form an equation using the doubling rule and solve it.
Angles in the same segment, cyclic quadrilaterals, tangents, the alternate segment theorem and radius-tangent theorem.
