Circle Theorems GCSE Maths
Circle theorems are a common Higher GCSE topic. Many students find them difficult at first because there are several rules to remember, but the topic becomes much easier when each theorem is learned clearly and practised regularly.
What Are Circle Theorems?
Circle theorems are rules about angles and lines inside circles.
In exams, students usually need to:
- remember the theorem
- spot which theorem applies
- use it together with other angle facts
Main Circle Theorems to Know
1. The Angle at the Centre Is Twice the Angle at the Circumference
If two angles stand on the same arc, the angle at the centre is double the angle at the circumference.
Example: if the angle at the circumference is 35°, then the angle at the centre is 70°.
2. Angles in the Same Segment Are Equal
If two angles stand on the same chord, they are equal.
This is one of the most common theorems in GCSE exam questions.
3. The Angle in a Semicircle Is 90°
If a triangle is drawn inside a semicircle and one side is the diameter, the angle opposite the diameter is always 90°.
4. Opposite Angles in a Cyclic Quadrilateral Sum to 180°
A cyclic quadrilateral is a four-sided shape with all corners on the circle.
Opposite angles add to 180°.
5. The Tangent Is Perpendicular to the Radius
A tangent touches the circle at one point only. The radius to that point makes a 90° angle with the tangent.
6. Tangents from the Same External Point Are Equal in Length
If two tangents are drawn from the same point outside the circle, they have equal length.
7. The Alternate Segment Theorem
The angle between a tangent and a chord equals the angle in the opposite segment.
Many students find this theorem one of the hardest, so it is important to practise recognising the diagram.
How to Answer Circle Theorem Questions
- look carefully at the diagram
- identify any tangent, radius, chord, diameter, or cyclic quadrilateral
- spot which theorem applies
- combine it with basic angle facts if needed
- write the theorem name if the question asks for a reason
Example
If an angle at the circumference standing on an arc is 28°, what is the angle at the centre standing on the same arc?
By the circle theorem, the angle at the centre is twice the angle at the circumference.
2 × 28° = 56°
Common Mistakes
- mixing up similar-looking theorems
- forgetting that a tangent and radius make 90°
- using a theorem that does not match the diagram
- not giving a reason when the exam asks for one
- forgetting to combine circle theorems with normal angle facts
Revision Tip
Circle theorems are best revised by drawing small labelled diagrams and testing yourself on what rule applies to each one. Memorising words alone is usually not enough.