IGCSE Maths / Circle theorems
The angle at the centre is twice the angle at the circumference
Edexcel International GCSE Maths (4MA1), chapter 17: Circle theorems, specification point 4.6c. Free lesson.
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When two angles stand on the same arc, the angle at the centre is twice the angle at the circumference. This lets you find either angle and explain why your answer is correct.
The rule for angles on the same arc
An arc is a piece of the circle’s edge. An angle is said to stand on an arc when its two lines end at the arc’s endpoints. The centre angle is twice the circumference angle only when both angles stand on the same arc. ∠ AOB = 2 × ∠ APB, when both angles stand on arc AB.
Find the angle at the centre
When you know the angle at the circumference, double it to find the angle at the centre. Check the endpoints first: both angles must stand on the same pair of points.
Only use angles standing on the same arc
You cannot double just any angle on the circle. The two lines from the centre and the two lines from the circumference point must all end at the same two points. If they end at a different pair, the rule does not connect those angles. The same-arc condition is part of the theorem, not an optional check.
Work back from the centre
The centre angle is the bigger angle in the pair. So, when you are given the centre angle and need the circumference angle, halve it. A quick check is that doubling your answer should give the centre angle you started with.
Give the reason in full
A correct number is not the whole answer when the question asks for a reason. State the theorem clearly and include the same-arc condition. Use: “The angle at the centre is twice the angle at the circumference, as both angles stand on the same arc.”
More lessons in Circle theorems
- Parts of a circle and cyclic quadrilaterals
- Tangent and radius, and two tangents from a point
- The angle in a semicircle
- Angles in the same segment
- Opposite angles of a cyclic quadrilateral
- The alternate segment theorem
- Using the alternate segment theorem in a reasoning chain
- Reasoning chains with circle theorems
- Multi-step circle theorem problems
- Intersecting chords inside a circle
- Intersecting chords from a point outside the circle