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 circle has centre O. Radii OA and OB are drawn, together with the arc AB, enclosing a sector. The straight chord AB is also drawn. Write down the name of the region enclosed between the chord AB and the arc AB.
- 2.A, B, C and D are points on a circle, with B and D both on the same major arc AC. E lies on the straight line through A and B, beyond B, so that angle CBE = 145°. Work out the size of angle ADC.
- 3.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.
- 4.A straight line crosses a pair of parallel lines. One of the co-interior (allied) angles is 118°. Work out the size of the other co-interior angle.
- 5.In triangle ABC the angle at A and the angle at C are equal. The side AB is extended beyond B, and the exterior angle formed at B measures 98°. Work out the size of the angle at A.
- 6.A circle has centre O and radius 13 cm. AB is a chord of the circle, and M is the midpoint of AB. OM is drawn, and OM = 5 cm. Work out the length of AB.
- 7.Every diameter of a circle is an example of a more general type of straight line joining two points on the circle. Which term names that more general type of line?
- 8.Two circles are drawn with the same centre but different radii. Write down the term used to describe this pair of circles.
- 9.From an external point P, two tangents PA and PB touch a circle with centre O at A and B. Angle AOB = 112°. Work out the size of angle APB.
- 10.PT is a tangent to a circle with centre O, touching the circle at T. OT is a radius. Angle OPT = 27°, where P is a point outside the circle. Work out the size of angle POT.
- 11.A chord divides a circle into two segments of different sizes. Write down the name given to the smaller of the two segments.
- 12.In triangle ABC, angle A is 2x°, angle B is 3x° and angle C is 4x°. Work out the size of angle B.
- 13.A student is proving that opposite angles of a cyclic quadrilateral ABCD sum to 180°, letting a = angle DAB and c = angle BCD, and using the fact that the angle at the centre is twice the angle at the circumference. She has already written: 'By the angle at the centre theorem, the reflex angle BOD equals 2a and the non-reflex angle BOD equals 2c.' Which statement must come immediately after this one in a correct proof?
- 14.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.
- 15.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.
- 16.A regular polygon has 12 sides. Work out the size of one exterior angle of the polygon.
- 17.AB is a diameter of a circle with centre O, and C is a point on the circle. A student writes four statements to prove that angle ACB = 90°. Statement 1: OA = OC, since both are radii, so triangle OAC is isosceles with angle OAC = angle OCA. Statement 2: OB = OC, since both are radii, so triangle OBC is isosceles with angle OBC = angle OCB. Statement 3: in triangle ABC, the three angles sum to 360°, so angle OAC + angle OBC + angle ACB = 360°, meaning 2 × angle ACB = 360° and angle ACB = 180°. Statement 4: A, O and B lie on a straight line, since AB is a diameter through the centre O. Which one of these four statements is mathematically incorrect?
- 18.A circle has diameter 10 cm. Using π = 3.14, work out the circumference of the circle.
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.
Similar worksheets worth a look
- 🧮 Paper 1 non-calculator warm-up — Higher · 20 questions · ~25 min
- 📈 Quadratics: factorise, complete the square, formula · 24 questions · ~45 min
- ⚗️ Ratio and proportion mastery — Higher · 24 questions · ~45 min
- 🔺 Higher stretch — grade 8 and 9 topics · 18 questions · ~50 min