Centre and Circumference
The angle at the centre is twice the angle at the circumference standing on the same arc.
Learning Objectives
- use the angle-at-centre theorem;
- use the angle-in-a-semicircle theorem;
- use equal angles in the same segment;
- use opposite angles of cyclic quadrilaterals;
- apply tangent properties.
Introduction
This lesson develops Circle Theorems from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Main Circle Theorems
Semicircle
The angle in a semicircle is 90°.
Same Segment
Angles in the same segment are equal.
Cyclic Quadrilateral
Opposite angles add to 180°.
Radius and Tangent
A radius is perpendicular to the tangent at the point of contact.
Tangents from One Point
Tangent lengths from the same external point are equal.
Worked Examples
Angle at centre=124°. Find angle at circumference.
124÷2=62°.
One angle of a cyclic quadrilateral is 108°. Find the opposite angle.
180−108=72°.
PA and PB are tangents from P. PA=9 cm. Find PB.
PB=9 cm.
Try It: Angle at the Centre
For the same arc, the angle at the centre is twice the angle at the circumference.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
- The angle at the centre is 150°. Find the angle at the circumference.
- An angle in a semicircle is marked x. Find x.
- Opposite angles of a cyclic quadrilateral are x and 3x. Find x.
- A radius meets a tangent. Find the angle between them.
- Two tangents from P have lengths 4x+1 and 6x−9. Find x.
✅ Check Your Work
Show answers and working
- 75°.
- 90°.
- 4x=180, so x=45°.
- 90°.
- 4x+1=6x−9, so x=5.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Centre angle is twice circumference angle.
- Semicircle angle=90°.
- Cyclic opposite angles total 180°.