Circle theorems
Circle theorem questions give you a diagram and ask for an angle with reasons. Most marks come from naming the theorem you used, so learn the wording as well as the picture.
Circle theorems are on the Higher tier papers for AQA, Edexcel and OCR.
Part 1 of 3: Learn it
In short
- Look for radii first: two radii make an isosceles triangle.
- Every step needs a reason, in words, after the angle.
- A cyclic quadrilateral has all four corners on the circle; opposite angles add to 180°.
Where this is in your specification
Spec points: DfE G10 (Higher tier)
| Board | Topic: Circles and circle theorems |
|---|---|
| DfE content | G9, G10, A16 |
| AQA 8300 | Geometry and measures; Algebra |
| Edexcel 1MA1 | Geometry and measures; Algebra |
| OCR J560 | 8 Basic geometry; 6 Graphs of equations and functions |
The theorems
| Theorem | What to look for |
|---|---|
| The angle at the centre is twice the angle at the circumference. | two chords from the ends of an arc meet at the centre and at the edge |
| The angle in a semicircle is 90°. | a triangle with one side as a diameter |
| Angles in the same segment are equal. | two angles at the edge standing on the same chord, on the same side |
| Opposite angles in a cyclic quadrilateral add up to 180°. | a four-sided shape with every corner on the circle |
| A tangent is perpendicular to the radius. | a line touching the circle once, meeting a radius |
| Tangents from an external point are equal in length. | two tangents meeting outside the circle (a kite shape with the centre) |
| The perpendicular from the centre to a chord bisects the chord. | a line from the centre meeting a chord at 90° |
| The alternate segment theorem. | the angle between a tangent and a chord equals the angle in the other segment |
Writing reasons
Write the angle, then the reason in full: 'angle ABC = 90° because the angle in a semicircle is 90°'. Use ordinary angle facts too: angles in a triangle add to 180°, angles on a straight line add to 180°, base angles of an isosceles triangle are equal.
A triangle is drawn with one side a diameter. What is the angle opposite the diameter?
Show the answer
90°: the angle in a semicircle is 90°.
Part 2 of 3: See it worked
Worked examples
Example 1
The four corners of PQRS all lie on a circle. Angle SPQ = 75° and angle PQR = 110°. Find angles QRS and RSP, with reasons.
- Angle QRS is opposite angle SPQ: 180° − 75° = 105°
- Reason: opposite angles in a cyclic quadrilateral add up to 180°
- Angle RSP is opposite angle PQR: 180° − 110° = 70°
- Check: 75 + 110 + 105 + 70 = 360°
Answer: Angle QRS = 105° and angle RSP = 70°, because opposite angles in a cyclic quadrilateral add up to 180°.
Example 2
O is the centre. Points A, B and C are on the circle and angle AOC = 130°. Find angle ABC, where B is on the major arc.
- Angle AOC is at the centre; angle ABC is at the circumference on the same arc
- Angle ABC = 130° ÷ 2
Answer: 65°, because the angle at the centre is twice the angle at the circumference.
Common mistakes
- Giving an angle with no reason, or a reason like 'circle theorem' without saying which one.
- Using the cyclic quadrilateral rule when one corner is the centre, not on the circle.
- Doubling instead of halving with the centre theorem.
- Assuming a line is a diameter because it looks like one. It must pass through the centre.
A tangent meets a radius. What is the angle between them?
Show the answer
90°: a tangent is perpendicular to the radius.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
A triangle is drawn with one side a diameter. What is the angle opposite the diameter?
90°: the angle in a semicircle is 90°.
A tangent meets a radius. What is the angle between them?
90°: a tangent is perpendicular to the radius.
Two angles stand on the same chord in the same segment. One is 42°. What is the other?
42°: angles in the same segment are equal.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses