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.Two circles are drawn with the same centre but different radii. Write down the term used to describe this pair of circles.
- 2.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°?
- 3.ABCD is a cyclic quadrilateral, with its vertices in that order around the circle. Angle DAB = 108°. Work out the size of angle BCD.
- 4.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.
- 5.A regular polygon has an exterior angle of 45°. Work out the number of sides of the polygon.
- 6.A, B and C are points on a circle with centre O. B is on the major arc AC. Angle AOC = 116°. Work out the size of angle ABC.
- 7.A regular polygon has 12 sides. Work out the size of one exterior angle of the polygon.
- 8.Points A, B, C and D lie on a circle with centre O, in that order around the circle, and AC is a diameter. Which of the following circle facts does NOT apply to this diagram?
- 9.A circular table mat has diameter 20 cm. Yasmin has a 70 cm length of ribbon and sews it around the edge of the mat with no overlap. Using π = 3.14, work out how much ribbon is left over after going around the mat once.
- 10.A tangent to a circle touches the circle at exactly one point, P. Work out the size of the angle between the tangent and the radius drawn to P.
- 11.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.
- 12.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.
- 13.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.
- 14.A circle has diameter 10 cm. Using π = 3.14, work out the circumference of the circle.
- 15.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.
- 16.A circle with centre O has radius 17 cm. A chord AB is drawn so that the perpendicular distance from O to AB is 8 cm. Work out the length of the chord AB.
- 17.Two identical ladders lean against the same vertical wall from opposite sides, each making an angle of 58.2° with the ground. The two ladders and the ground form a triangle. Work out the size of the angle between the two ladders at the top, where they meet.
- 18.A circle has circumference 31.4 cm. Using π = 3.14, work out the radius 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