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 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.
- 2.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.
- 3.A sector of a circle has an angle of 90° at the centre. Write down what fraction of the whole circle this sector represents.
- 4.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.
- 5.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.
- 6.A tangent to a circle with centre O touches the circle at point P, where OP = 5 cm. Point Q lies on the tangent so that PQ = 12 cm. Using the fact that a tangent is perpendicular to the radius at the point of contact, work out the length OQ.
- 7.To prove that the angle at the centre is twice the angle at the circumference, a student draws radii OA and OC, where A and C are points on the circle and O is the centre. The student's first step is: 'Since OA = OC, both being radii of the circle, triangle OAC is isosceles, so angle OAC = angle OCA.' Is this first step correct?
- 8.ABCD is a cyclic quadrilateral, with its vertices in that order around the circle. Angle DAB = 108°. Work out the size of angle BCD.
- 9.Four angles meet at a point. Three of them measure 82°, 105° and 96°. Work out the size of the fourth angle.
- 10.A, B and C are points on a circle with centre O, and AC is a diameter of the circle. Work out the size of angle ABC.
- 11.A, B and C are points on a circle with centre O. B is on the major arc AC. Angle AOC = (4x + 20)°. Work out the size of angle ABC in terms of x, in its simplest form.
- 12.Two points on a circle divide its circumference into two arcs of different lengths. Write down the name given to the shorter of the two arcs.
- 13.A ramp's sloped surface crosses two horizontal parallel rails. At the top rail, the angle between the ramp and the rail on the right of the ramp is (3x + 10)°. At the bottom rail, the angle between the ramp and the rail on the left of the ramp is (5x − 30)°. Work out the value of x.
- 14.AT is a diameter of a circle with centre O. PT is a tangent to the circle at the point T, and P is a point on this tangent such that A, T and P form a triangle. Angle PAT = 28°. Work out the size of angle APT.
- 15.A circle has two straight radii drawn from the centre to the edge, together with the arc between them. Write down the name of the region enclosed by these two radii and the arc.
- 16.A conservatory has a roof panel shaped like a regular octagon. Work out the interior angle of the octagon, then use it to work out the size of the reflex angle at the same vertex, on the outside of the roof panel.
- 17.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.
- 18.A regular polygon has an interior angle of 156°. Work out the number of sides of the polygon.
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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