, and are points on a circle with centre . Angle , and is on the major arc.
- Work out angle . Give a reason for your answer.[2]
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Draw any triangle with its three corners on a circle and the diameter as one side, and the angle opposite the diameter is always . That is one of a set of circle theorems — reliable angle facts that let you find unknown angles and prove geometric results. AQA expects you to apply them with correct reasons and, at Higher tier, to prove them.
The big picture
Most circle theorem questions are angle-chasing: find one angle, use it to find the next, and write a reason for each step. The theorems sit alongside ordinary angle facts — angles in a triangle, isosceles triangles, angles on a straight line.
Two radii always make an isosceles triangle. Spotting those triangles is often the step that unlocks the whole problem, and it is also the key to proving the theorems.
What you'll learn
1. The angle at the centre is twice the angle at the circumference subtended by the same arc.
2. The angle in a semicircle is .
3. Angles in the same segment are equal.
4. Opposite angles of a cyclic quadrilateral add up to .
An arc subtends an angle of at the centre of a circle. What angle does it subtend at the circumference (on the major arc)?
In a cyclic quadrilateral, one angle is . Find the opposite angle.
Tip — Check that a quadrilateral really is cyclic — all four vertices must lie on the circle, and the centre must not be one of them.
5. A tangent meets a radius at at the point of contact.
6. Tangents from an external point are equal in length.
7. The perpendicular from the centre to a chord bisects the chord.
8. The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
A circle has radius 8 cm. A point is 17 cm from the centre. Find the length of a tangent from to the circle, in cm.
A tangent and radius create a right angle, so many circle problems become Pythagoras or trigonometry once that angle is marked.
Use the standard wording. "Angle at centre is twice angle at circumference" earns the mark; "because of the circle rule" does not.
Give a reason for every angle you find, including ordinary facts like "angles in a triangle add up to " and "base angles of an isosceles triangle are equal".
Tip — Mark each angle on the diagram as you find it. It keeps the chain of reasoning clear for you and the examiner.
To prove the angle in a semicircle is : let be a diameter with centre and on the circle.
(radii), so triangles and are isosceles. Let angle and angle ; then angle and angle .
The angles of triangle add to : , so , which is angle .
Think like an examiner
Remember these
Watch out for these
Stretch yourself
Prove that opposite angles of a cyclic quadrilateral add up to . You may use the fact that the angle at the centre is twice the angle at the circumference.
Hint — Join both remaining vertices to the centre. The two angles at the centre make a full turn.
Questions students ask
Key takeaways
Exam-style questions
3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.
, and are points on a circle with centre . Angle , and is on the major arc.
is a tangent to a circle with centre , touching it at . Angle .
is a cyclic quadrilateral. Angle and angle .
Test yourself
Ready to practise Circle Theorems? Pick a mode and earn XP & Dobloons.
Real past-paper questions on Circle Theorems, marked mark-by-mark. How you do feeds straight into your weak-topic list, so your revision keeps targeting what actually needs work.