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Circle Theorems - Edexcel GCSE Higher Maths

Typically 4-6 marks per paper, usually one longer question with reasons · Spec 1MA1

Circle theorems are the angle facts that hold whenever chords, tangents and radii meet a circle. On Edexcel's 1MA1 specification they sit at reference G10, and unlike most geometry content G10 is Higher tier only, so a Foundation candidate never meets them. The list is short and worth writing out from memory: the angle at the centre is twice the angle at the circumference standing on the same arc, the angle in a semicircle is 90∘90^\circ, angles in the same segment are equal, opposite angles of a cyclic quadrilateral add to 180∘180^\circ, a tangent meets a radius at 90∘90^\circ, the two tangents from a point outside the circle are equal in length, a perpendicular from the centre bisects a chord, and the alternate segment theorem.

Edexcel can set G10 on any of the three papers, and the calculator makes little difference, because the arithmetic is subtraction, halving and the occasional application of Pythagoras. What changes the question is the instruction attached to it: 'Give a reason for each stage of your working'. Those versions run to 4-5 marks and one of them is a communication mark awarded for the wording alone. Every figure is printed with 'Diagram NOT accurately drawn', so an angle that looks square is not square until a theorem says so, and two chords that look the same length are not equal until you can name why.

The geometry is rarely what costs the marks. The same habits cost them instead: a candidate writes 140 in the working with nothing to say which angle it is, or quotes a theorem loosely as 'the angle at the origin is twice the angle at the edge', or invents an isosceles triangle because two lines look alike. Write every angle you find in three-letter notation, so angle BCD = 65∘65^\circ rather than a lone 65. Quote each theorem in full, condition included. And remember which isosceles triangle you are always entitled to: any two radii are equal, so a triangle with the centre as one vertex and two points of the circle as the others has two equal base angles.

Worked example

A, B and C are points on the circumference of a circle with centre O. O lies inside triangle ABC. Angle OAB = 32∘32^\circ and angle OCB = 21∘21^\circ. Work out the size of angle AOC. Give a reason for each stage of your working.
[4 marks]
  1. OA and OB are radii, so OA = OB and triangle OAB is isosceles. Its base angles are equal, so angle OBA = 32∘32^\circ. In the same way OB = OC, so angle OBC = angle OCB = 21∘21^\circ.
    M1: both isosceles triangles used, justified by equal radii rather than by how the lines look
  2. O is inside the triangle, so the ray BO lies between BA and BC: angle ABC = 32+21=53∘32 + 21 = 53^\circ.
    M1: combining the two base angles into the full angle at B
  3. Angle AOC and angle ABC both stand on the arc AC, so angle AOC = 2×53=106∘2 \times 53 = 106^\circ.
    A1: 106, doubling rather than halving; the centre angle is the larger of the pair
  4. Reasons, written out: base angles of an isosceles triangle are equal (OA = OB = OC as radii); the angle at the centre is twice the angle at the circumference when both stand on the same arc. Final line: angle AOC = 106∘106^\circ.
    C1: the communication mark, for correctly named theorems and for tying 106 to angle AOC by name

Practice questions

These are original questions in Edexcel 1MA1 Higher style. There are no diagrams, so every figure is spelled out: which points lie on the circle and in what order, which line is the tangent and where it touches, and which chord is a diameter. Sketch each one before you calculate, then name the theorem you used, because on the real paper one of the marks is for the wording rather than the number.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Warm-up [1 mark]
    A, B and C are points on the circumference of a circle with centre O. B lies on the major arc AC. Angle ABC = 34∘34^\circ. Work out the size of angle AOC, in degrees.
  2. Question 2 Warm-up [2 marks]
    T is a point on the circumference of a circle with centre O. The line PT is a tangent to the circle, touching it at T. Angle OPT = 32∘32^\circ. Work out the size of angle POT, in degrees.
  3. Question 3 Exam pace [1 mark]
    T is a point on a circle with centre O, and the line VT is a tangent to the circle at T. A student writes angle OTV = 90∘90^\circ. Which wording would be accepted as the reason?
    Answer options for question 3
  4. Question 4 Exam pace [1 mark]
    A, B and C are points on a circle. The line SBT is a tangent to the circle at B, with S and T on opposite sides of B and with S and C on opposite sides of the line AB. A student writes angle ABS = angle ACB. Which reason should be given?
    Answer options for question 4
  5. Question 5 Exam pace [3 marks]
    A circular stained glass window has centre O and radius 40 cm. A straight lead strip runs across the window from one point on the edge to another, and the strip is 64 cm long. Work out the shortest distance from O to the strip, in cm.
  6. Question 6 Stretch [4 marks]
    A, B and C are points on the circumference of a circle. The line SBT is a tangent to the circle at B, with S and T on opposite sides of B. Angle ABS = 52∘52^\circ and angle CBT = 61∘61^\circ. D is a point on the arc AC that does not contain B. Work out the size of angle BAC and the size of angle ADC, in degrees.

Common mistakes examiners see

  • Running the centre theorem backwards. Given an angle of 34∘34^\circ at the circumference, halving it to 17∘17^\circ instead of doubling it to 68∘68^\circ at the centre.

    Say which angle is which before you calculate: the one at O is always the bigger of the pair. A quick sketch check is that a 90∘90^\circ angle at the circumference sits on a diameter, which is 180∘180^\circ at the centre.

  • Using neighbouring angles of a cyclic quadrilateral. In ABCD, subtracting angle ABC from 180∘180^\circ to get angle BCD, when the rule only links angle ABC with angle ADC.

    Mark the two diagonals lightly before you start. The pairs that add to 180∘180^\circ are the ends of each diagonal's opposite corners, so A pairs with C and B pairs with D. Adjacent angles have no fixed sum at all.

  • Finding the right angle and losing the mark for the reason, or writing something close but not accurate, such as 'the angle between a tangent and the circle is 90∘90^\circ' or 'angles in a circle add up to 180∘180^\circ'.

    Learn the eight statements word for word and write the full sentence beside each line of working, including the condition. The right angle is between the tangent and the radius drawn to the point of contact, and the centre theorem needs the phrase 'standing on the same arc'.

  • Applying the alternate segment theorem to the wrong vertex, so the tangent-chord angle is set equal to the triangle angle sitting next to the chord instead of the one facing it across the circle.

    Trace the chord from the point of contact and find the angle at the far vertex, the one the chord points away from. In triangle ABC with the tangent at B, the angle between the tangent and BA equals angle ACB, not angle BAC.

  • Assuming what the diagram looks like: treating OC and BC as equal because they are drawn about the same length, or calling a line a tangent because it grazes the edge of the figure.

    Only use equalities the question states or a theorem gives you: two radii, two tangents from the same external point, or the two halves of a chord cut by a perpendicular from the centre. If nothing on that list applies, the triangle is not isosceles.

Frequently asked questions

Are circle theorems on the Foundation paper?
No. G10 is one of the Higher-only spec references on Edexcel 1MA1, so circle theorems appear on Papers 1, 2 and 3 at Higher tier and nowhere at Foundation. Foundation candidates still meet circle vocabulary such as chord, arc and tangent, and the area and circumference formulae, but not the angle theorems.
How many circle theorems do I need to learn for Edexcel GCSE Higher?
Eight cover the specification: angle at the centre is twice the angle at the circumference, angle in a semicircle is a right angle, angles in the same segment are equal, opposite angles of a cyclic quadrilateral add to 180 degrees, tangent meets radius at 90 degrees, two tangents from a point are equal in length, a perpendicular from the centre bisects a chord, and the alternate segment theorem. None of them is printed on the formulae sheet.
Do I lose marks if I get the angle right but do not give a reason?
Yes, when the question says 'give a reason' or 'you must give reasons for each stage of your working'. Edexcel puts a separate mark on the reasoning, and it is awarded for a correctly worded theorem, not for a vague description. Naming the theorem for every angle you find, not only the last one, is the safest habit.
Does the alternate segment theorem come up often on Edexcel papers?
It appears regularly, and it is the theorem students reach for last, going the long way round through the centre instead. Look for it whenever a tangent and a chord meet at a point on the circle: the angle between them equals the angle subtended by that chord from the opposite arc.

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