Circle theorems are a Higher tier topic tested across AQA, Edexcel and OCR — and they are consistently worth 4–8 marks per paper. The challenge is not the maths itself (which is usually just angle arithmetic once you know the theorem) but knowing which theorem applies and being able to name it when required.
This guide covers all eight circle theorems with clear explanations, worked examples and the exact wording needed for exam marks.
Before You Start: Key Vocabulary
Chord: A straight line connecting two points on the circumference.
Arc: A portion of the circumference between two points.
Tangent: A straight line that touches the circle at exactly one point without crossing it.
Radius (plural: radii): A straight line from the centre to the circumference.
Diameter: A chord that passes through the centre (= 2 × radius).
Subtend: An angle is said to be "subtended" by an arc or chord when the arc/chord is opposite the angle.
Cyclic quadrilateral: A quadrilateral with all four vertices on the circumference of a circle.
Theorem 1: Angle at the Centre
The angle at the centre of a circle is twice the angle at the circumference, when both are subtended by the same arc.
In other words: if you have an arc AB, and you draw a line from A and B to the centre O to form angle AOB, and also draw lines from A and B to any point P on the major arc to form angle APB — then angle AOB = 2 × angle APB.
Worked Example 1
O is the centre of a circle. Angle AOB = 130°. P is a point on the major arc. Find angle APB.
Angle APB = ½ × 130° = 65°
Worked Example 2
O is the centre of a circle. Angle APB = 48°. Find the reflex angle AOB.
Angle AOB (non-reflex) = 2 × 48° = 96°
Reflex angle AOB = 360° − 96° = 264°
Theorem 2: Angle in a Semicircle
The angle in a semicircle is always 90°.
More precisely: if AB is a diameter, then any angle ACB where C is on the circumference (and not on the diameter) is always 90°.
This is actually a special case of Theorem 1 — the diameter subtends 180° at the centre, so it subtends 90° at the circumference.
Worked Example 3
AB is a diameter of a circle. C is a point on the circumference. Angle CAB = 38°. Find angle ABC.
Since angle ACB = 90° (angle in semicircle), the angles in triangle ACB sum to 180°.
Angle ABC = 180° − 90° − 38° = 52°
On the exam: Always state "angle in a semicircle = 90°" when using this theorem.
Theorem 3: Angles in the Same Segment
Angles subtended by the same chord at the same side of the chord are equal.
If two or more points lie on the same arc (same side of chord AB), then all angles subtended at the circumference by chord AB are equal.
Worked Example 4
A, B, C and D are points on a circle. AC and BD are chords that cross at E. Angle AEB = 65°. Find angle ADB.
Angles CAB and CDB are in the same segment (both subtended by arc CB) — so they are equal.
If you know angle CAB = 40°, then angle CDB = 40° also.
Theorem 4: Cyclic Quadrilateral
Opposite angles in a cyclic quadrilateral sum to 180°.
A cyclic quadrilateral is a four-sided shape with all four corners on the circumference. The two pairs of opposite angles each add up to 180°.
So if angles are A, B, C and D (going around the quadrilateral):
- A + C = 180°
- B + D = 180°
Worked Example 5
ABCD is a cyclic quadrilateral. Angle A = 85°, angle B = 110°. Find angles C and D.
Angle C = 180° − 85° = 95°
Angle D = 180° − 110° = 70°
Check: A + B + C + D = 85 + 110 + 95 + 70 = 360° ✓
Theorem 5: Tangent-Radius
A tangent to a circle is perpendicular to the radius at the point of contact.
If a tangent touches the circle at point T, and O is the centre, then the angle OTP = 90° (where TP is the tangent).
Worked Example 6
A tangent touches a circle at T. O is the centre. A line from O to a point P on the tangent makes an angle of 35° with OT. Find angle OPT.
Triangle OTP has a right angle at T (tangent-radius theorem).
Angle OTP = 90°, angle TOP = 35°
Angle OPT = 180° − 90° − 35° = 55°
Theorem 6: Two Tangents from an External Point
Tangents drawn from an external point to a circle are equal in length.
If tangents are drawn from an external point P to the circle, touching at A and B, then PA = PB. The line from P to the centre O bisects the angle at P and the angle at O.
Worked Example 7
Two tangents from point P touch a circle at A and B. PA = 3x + 2 cm and PB = 5x − 6 cm. Find the length of each tangent.
Since tangents from an external point are equal: 3x + 2 = 5x − 6 → 8 = 2x → x = 4
PA = PB = 3(4) + 2 = 14 cm
Theorem 7: Alternate Segment Theorem
The angle between a tangent and a chord equals the angle in the alternate segment.
This is the most conceptually difficult theorem. When a chord touches the circumference at a point on a tangent, the angle between the chord and the tangent equals the angle subtended by the chord in the opposite (alternate) segment of the circle.
In practice: if you have a tangent at point A, and a chord AB, the angle between the tangent and AB on one side equals the inscribed angle ACB on the other side (where C is any point in the opposite arc).
Worked Example 8
A tangent to a circle at point A makes an angle of 63° with chord AB. C is a point in the alternate segment. Find angle ACB.
By the alternate segment theorem: angle ACB = 63°
Worked Example 9 (combination)
A tangent at A makes an angle of 50° with chord AB. Chord BC subtends an angle at the circumference. Find angle ABC in triangle ABC if angle ACB = 50° and angle BAC = 68°.
Angle ABC = 180° − 50° − 68° = 62°
Theorem 8: Perpendicular from Centre to Chord
The perpendicular from the centre of a circle to a chord bisects the chord.
If O is the centre and you draw a perpendicular from O to chord AB, the perpendicular meets AB at its midpoint M (so AM = MB). This creates two right-angled triangles.
Worked Example 10
A chord AB is 10 cm long. The perpendicular distance from the centre O to the chord is 5 cm. Find the radius of the circle.
The perpendicular bisects AB, so AM = 5 cm.
Using Pythagoras: r² = 5² + 5² = 25 + 25 = 50 → r = √50 = 5√2 cm ≈ 7.07 cm
How Exam Questions Combine Theorems
Exam questions frequently require you to use two or more theorems in sequence. A typical Higher tier question gives a diagram with several angles labelled and asks you to find an unknown angle, justifying each step.
Worked Example 11 — Multi-step
O is the centre of a circle. A, B and C are points on the circumference. Angle ACB = 34°. Calculate angle OAB. Give reasons.
Step 1: Angle AOB = 2 × angle ACB = 68° (angle at centre = twice angle at circumference)
Step 2: OA = OB (both radii), so triangle OAB is isosceles.
Step 3: Angle OAB = (180° − 68°) / 2 = 112° / 2 = 56°
For the reasons: each step needs a named theorem. "Angle at centre is twice angle at circumference" and "base angles of an isosceles triangle are equal" are the required justifications.
The 8 Circle Theorems — Quick Reference
| Theorem | Statement |
|---|---|
| 1. Angle at centre | Centre angle = 2 × circumference angle (same arc) |
| 2. Angle in semicircle | Angle in semicircle = 90° |
| 3. Same segment | Angles in same segment are equal |
| 4. Cyclic quadrilateral | Opposite angles sum to 180° |
| 5. Tangent-radius | Tangent ⊥ radius at point of contact |
| 6. Two tangents | Tangents from external point are equal in length |
| 7. Alternate segment | Tangent-chord angle = angle in alternate segment |
| 8. Perpendicular chord | Perpendicular from centre bisects the chord |
Exam Tips for Circle Theorems
State the theorem name. On AQA, Edexcel and OCR, marks for reasoning are only awarded if you name the theorem used. "Because of circle theorems" earns nothing. "Angle in a semicircle = 90°" earns the mark.
Mark all the radii. At the start of any circle theorem question, mark every radius you can see — it often reveals an isosceles triangle which unlocks the solution.
Look for isosceles triangles. Two radii of equal length form the equal sides of an isosceles triangle. This means the base angles are equal — frequently the key step in multi-theorem problems.
Draw the diagram clearly. If the exam diagram is small or cramped, redraw it larger in your working space. Circle theorem questions are much easier when you can see the geometry clearly.
Practise circle theorem questions with instant AI feedback — select Geometry & Measures → Circle Theorems for Higher tier practice questions.
Put this into practice — try AI-marked questions on this topic, completely free.
Start practising free →Test Yourself
Want to try a quick quiz on Circle Theorems?