18 questions on the standard circle theorems, each asking for the answer and the reason — as the exam does.
⭕ Circle theorems practice
Circle theorems are Higher tier only, and they are marked as much on the reason as on the number. This sheet asks for both, every time. It covers the angle at the centre being twice the angle at the circumference, the angle in a semicircle, angles in the same segment, opposite angles of a cyclic quadrilateral summing to 180°, the tangent meeting the radius at a right angle, the two tangents from a point being equal, the perpendicular from the centre bisecting a chord, and the alternate segment theorem. The later questions chain two theorems together, which is how they usually arrive on a paper. Write the theorem out in words rather than pointing at the diagram — 'angles in the same segment are equal' is the phrase that earns the mark, and it is the phrase that is easiest to leave out under time pressure.
- 1.A, B and C are points on a circle with centre O. B is on the minor arc AC. Angle ABC = 100°. Work out the size of the non-reflex angle AOC.
- 2.A tangent touches a circle with centre O at the point P. Q is a point on the tangent. Write down the circle fact that tells you the size of angle OPQ.
- 3.A circle has centre O. PQ, RS and TU are three chords of the circle, and only one of them passes through O. Write down what this tells you about that chord.
- 4.A circle has circumference 31.4 cm. Using π = 3.14, work out the radius of the circle.
- 5.A chord divides a circle into two regions, one larger than the other. Write down the term used for the smaller of these two regions.
- 6.A sector of a circle has an angle of 90° at the centre. Write down what fraction of the whole circle this sector represents.
- 7.A, B, C and D are points on a circle, and ABCD is a cyclic quadrilateral with the vertices in that order around the circle. Angle ABC = 108° and angle ADC = 72°. Which circle theorem is the reason that angle ABC + angle ADC = 180°?
- 8.ABCD is a cyclic quadrilateral, with its vertices in that order around the circle. Angle DAB = (2x + 10)° and angle DCB = (3x − 5)°. Work out the size of angle DAB.
- 9.A chord divides a circle into two segments of different sizes. Write down the name given to the smaller of the two segments.
- 10.A straight line touches the edge of a circle at exactly one point and does not cross into the circle at all. Write down the term for this line.
- 11.A ferris wheel's circular frame has centre O. A support strut runs from O to a point A on the rim, and a second strut runs from O to point B on the rim, with angle AOB = 76°. A cabin is mounted at point C on the major arc AB, and D is a separate point on the minor arc AB. Work out the difference between angle ACB and angle ADB.
- 12.Two circles are drawn with the same centre but different radii. Write down the term used to describe this pair of circles.
- 13.From an external point P, two tangents PA and PB touch a circle with centre O at the points A and B. Angle APB = 50°. Work out the size of angle OAB.
- 14.A trundle wheel has a diameter of 0.5 m. It is rolled along the ground and makes 20 complete turns. Using π = 3.14, work out the total distance rolled, in metres.
- 15.A, B, C and D are points on a circle with centre O, in that order around the circle. B lies on the major arc AC, and angle AOC = 152°. In triangle ACD, angle ACD = 30°. Work out the size of angle DAC.
- 16.The interior angles of a pentagon are 100°, 110°, 120°, x° and x°. Work out the size of each of the two angles marked x°.
- 17.ABCD is a cyclic quadrilateral, with its vertices in that order around the circle. Angle DAB = 108°. Work out the size of angle BCD.
- 18.A regular nonagon has 9 sides. Work out the sum of its interior angles.
What is on this worksheet?
The sheet holds 18 questions drawn from the MathsUK bank — the content area covered: Geometry and measures (statements G10, G9, G3). It is pitched at GCSE Higher and takes about 40 minutes to work through in full. It is built for independent practice, with full answers at the end for self-marking.
How to use the sheet well
- Print it or open it on screen — both work. Printing is A4; the screen view fits phones and tablets.
- Do all 18 questions before checking — about 40 minutes is the guide, but there is no time pressure.
- Check the answers — press “Show answers” or print the answer page separately.
- Redo the questions you got wrong — twice as effective as doing 18 fresh ones.
- “New questions” — builds a fresh sheet on the same statements, so you can practise again without repeats.
Why this sheet helps
MathsUK worksheets use questions graded by difficulty and a fair spread of correct-answer positions (the answer is not always (a)) — so the student really has to think about each question rather than guess a pattern. Every question is tagged to a DfE content statement and checked before it enters the bank. The answers come with a step-by-step explanation, not just a value — so a wrong answer becomes a lesson.
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