Find missing angles in circles using the circle theorems, and name the theorem you used as your reason. It is Higher only, aimed at grade 6. Free questions with answers are on this page. Worked examples show each line before you try the questions.
The circle theorems are a short list of named facts: angle in a semicircle is 90∘; angle at the centre is twice the angle at the circumference on the same arc; angles in the same segment are equal; opposite angles in a cyclic quadrilateral sum to 180∘; the tangent is perpendicular to the radius; two tangents from a point are equal; alternate segment theorem. Alternate segment: the angle between tangent and chord equals the angle in the other segment.
Mark the diagram, pick the theorem that uses the marked facts, write the reason in the accepted wording, then solve. Two-step questions need two reasons. Equal radii make an isosceles triangle — that is allowed and often the missing step.
Angle in a semicircle is 90∘. Centre is twice the circumference on the same arc. Opposite angles of a cyclic quadrilateral sum to 180∘. Angles in the same segment are equal. Write the theorem name as the reason.
Objectives
Halve an angle at the centre to get the angle at the circumference.
Double an angle at the circumference to get the angle at the centre.
Use the semicircle, cyclic quadrilateral, and tangent-radius facts.
Key points
State each circle theorem in words and spot it in a diagram.
Combine a circle theorem with basic angle facts (triangle, isosceles, straight line, quadrilateral).
Give a reason for every step using the exact theorem name.
Recognise isosceles triangles formed by two radii.
Worked examples
Example 1
Angle at the centre is 80 degrees. Find the angle at the circumference.
2 marks
Identify the rule: angle at circumference is half the centre angle.
Divide centre angle by 2: 80 ÷ 2 = 40.
State the angle is 40 degrees.
Answer: 40
Common slips
Slip: Assume circumference angle equals centre angle. The angles are in a 1:2 ratio on the same arc.
Slip: Multiply centre angle by 2: 80 × 2 = 160. The circumference angle is always smaller.
Slip: Double it to 160 degrees. The circumference angle on the same arc is half.
Example 2
Angle at the circumference is 35 degrees. Find the angle at the centre.
2 marks
Identify the rule: angle at centre is double circumference angle.
Multiply circumference angle by 2: 35 × 2 = 70.
State the centre angle is 70 degrees.
Answer: 70
Common slips
Slip: Halve the circumference angle. The centre angle is twice as large.
Slip: Divide by 2: 35 ÷ 2 = 17.5. The centre angle subtends the same arc with twice the opening.
Slip: Write 17.5 degrees. The centre angle is 70 degrees.
Example 3
A cyclic quadrilateral has an angle of 70 degrees. Find the opposite angle.
3 marks
Recall that opposite angles in a cyclic quadrilateral sum to 180.
Subtract from 180: 180 - 70 = 110.
State the opposite angle is 110 degrees.
Answer: 110
Common slips
Slip: Assume opposite angles are equal. Opposite angles in a cyclic quadrilateral add to 180. They are only equal when both are 90.
Slip: Subtract from 90: 90 - 70 = 20. Opposite angles add to 180, not 90.
Slip: Write 70 degrees. The opposite angle is 110 degrees.
Questions and answers
GCSE Maths circle theorems questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Angle at the centre is 100 degrees. Find the angle at the circumference.
Show answer
Answer: 50
Circumference angle is half centre angle.
100 / 2 = 50.
M1 — Circumference angle is half centre angle.
A1 — 100 / 2 = 50.
Question 2 Easy2 marks
Angle at the centre is 120 degrees. Find the angle at the circumference.
Show answer
Answer: 60
Circumference angle is half centre angle.
120 / 2 = 60.
M1 — Circumference angle is half centre angle.
A1 — 120 / 2 = 60.
Question 3 Medium2 marks
Angle at the circumference is 40 degrees. Find the angle at the centre.
Show answer
Answer: 80
Centre angle is double circumference angle.
40 × 2 = 80.
M1 — Centre angle is double circumference angle.
A1 — 40 × 2 = 80.
Question 4 Medium2 marks
Angle at the circumference is 55 degrees. Find the angle at the centre.
Show answer
Answer: 110
Centre angle is double circumference angle.
55 × 2 = 110.
M1 — Centre angle is double circumference angle.
A1 — 55 × 2 = 110.
Question 5 Hard2 marks
Find the angle in a semicircle.
Show answer
Answer: 90
An angle subtended by a diameter at the circumference is 90 degrees.
The triangle in a semicircle has a right angle.
M1 — An angle subtended by a diameter at the circumference is 90 degrees.
A1 — The triangle in a semicircle has a right angle.
Question 6 Hard2 marks
Find the angle between a tangent and the radius.
Show answer
Answer: 90
A tangent meets a radius at right angles.
The angle is 90 degrees.
M1 — A tangent meets a radius at right angles.
A1 — The angle is 90 degrees.
Question 7 Exam-style4 marks
A, B, C and D are points on a circle with centre O. C is on the major arc AB and D is on the minor arc AB. Angle AOB is 140 degrees. Work out angle ADB. Give reasons for your working.
Show answer
Answer: 110
Angle ACB = 140 ÷ 2 = 70 degrees, because the angle at the centre is twice the angle at the circumference.
ACBD is a cyclic quadrilateral, so angle ADB = 180 - 70 = 110 degrees, because opposite angles add to 180.
M1 — 140 ÷ 2 = 70
M1 — 180 - 70
A1 — 110
B1 — Reasons given: angle at centre is twice angle at circumference, and opposite angles of a cyclic quadrilateral add to 180
Common mistakes
Most lost marks on this topic are one slip.
Doubling the centre angle to get the circumference angle
Halve the centre angle
Halving the circumference angle to get the centre angle
Double the circumference angle
Using 90 degrees for a cyclic opposite angle
Subtract the given angle from 180
Spot the slip
The angle at the centre is 80 degrees. Sue doubles it to find the angle at the circumference on the same arc. What is the angle at the circumference?
40
80
160
Show the slip
40 The angle at the circumference is half the angle at the centre on the same arc.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Reasons are marked separately from numbers. A correct angle with no theorem named usually loses a mark.
Write angles on the diagram as you find them, then copy the chain of reasoning underneath in order.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.