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.O is the centre of a circle, and PT is a tangent to the circle at the point T. A is a point on the circle, with OA and OT both radii and angle AOT = 122°. At the point T, the chord TA lies between the radius TO and the tangent TP. Work out the size of angle ATP, the angle between the chord and the tangent.
- 2.At a cycling event, a circular track has centre O. Two flags, F and M, are fixed on the track. A course marker P is also on the track, positioned so that angle FOP = 55° and angle POM = 43°, with P between F and M as seen from O. A spectator S stands on the major arc FM, on the opposite side of the track from P. The event programme needs angle FSM. Work out angle FSM.
- 3.A, B, C and D are points on a circle with centre O, placed in that order around the circle so that ABCD is a cyclic quadrilateral. B lies on the major arc AC and D lies on the minor arc AC. Angle AOC = 104°. Work out the size of angle ADC.
- 4.A circle has diameter 10 cm. Using π = 3.14, work out the circumference of the circle.
- 5.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.
- 6.In triangle ABC, angle A is 2x°, angle B is 3x° and angle C is 4x°. Work out the size of angle B.
- 7.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.
- 8.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.
- 9.A straight line crosses two parallel lines. One of the angles formed is (2x + 10)°, and the angle alternate to it is 74°. Work out the value of x.
- 10.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.
- 11.From an external point P, two tangents PA and PB touch a circle with centre O at A and B. Angle APB = 40°. Work out the size of angle AOB.
- 12.PT is a tangent to a circle with centre O, touching the circle at T. C is a point on the circle such that C lies inside angle OTP (the right angle between the radius OT and the tangent PT), and OC is a radius. Angle TOC = 130°. Work out the size of angle PTC, the angle between the tangent PT and the chord TC.
- 13.A circle has circumference 31.4 cm. Using π = 3.14, work out the radius of the circle.
- 14.The sum of the interior angles of a polygon is 1980°. Work out the number of sides of the polygon.
- 15.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.
- 16.A regular polygon has an exterior angle of 45°. Work out the number of sides of the polygon.
- 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.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°.
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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