Circle theorems are a set of rules about angles and lines in circles. They only appear on Higher tier papers, but once you know all seven, most questions become quick angle-chasing exercises.
1. Angle in a semicircle
The angle in a semicircle is always 90°. If AB is a diameter and C is any other point on the circle, angle ACB = 90°.
Worked Example 1
AB is a diameter. Angle BAC = 30°. Find angle ABC.
1
Angle ACB = 90° (angle in a semicircle)
2
Angles in a triangle sum to 180°: angle ABC = 180° − 90° − 30°
AnswerAngle ABC = 60°
AB is a diameter, so angle ACB = 90° regardless of where C sits on the circle.
2. Angle at the centre
The angle at the centre of a circle is twice the angle at the circumference, when both angles are subtended by the same arc.
Worked Example 2
Angle AOB (at the centre) = 140°. Find angle ACB (at the circumference), where C is on the major arc.
1
The angle at the centre is twice the angle at the circumference: angle ACB = angle AOB ÷ 2
AnswerAngle ACB = 70°
Angle AOB (at the centre, black) is exactly double angle ACB (at the circumference, teal) — both are subtended by arc AB.
3. Angles in the same segment
Angles subtended by the same arc, from points in the same segment, are equal. In the diagram below, angle ACB = angle ADB, because both are subtended by arc AB from the same side.
C and D are both on the same arc. The angles they see arc AB at (angle ACB and angle ADB) are equal.
4. Cyclic quadrilaterals
A cyclic quadrilateral has all four vertices on a circle. Opposite angles in a cyclic quadrilateral always sum to 180°.
Worked Example 3
ABCD is a cyclic quadrilateral. Angle A = 100°. Find angle C.
1
Opposite angles in a cyclic quadrilateral sum to 180°: angle C = 180° − angle A
AnswerAngle C = 80°
ABCD is a cyclic quadrilateral: angle A + angle C = 180°, and angle B + angle D = 180°.
5. Tangents and radii
A tangent touches a circle at exactly one point. At that point, the tangent is always perpendicular (at 90°) to the radius.
The tangent at A meets the radius OA at exactly 90°.
6. Two tangents from a point
If two tangents are drawn from the same external point to a circle, the two tangent lengths are equal. The line from the external point to the centre bisects the angle between the tangents.
Worked Example 4
O is the centre of a circle with radius 6cm. P is an external point with OP = 10cm. Tangents from P touch the circle at A and B. Find the length PA.
1
OA is a radius and PA is a tangent, so angle OAP = 90° (tangent perpendicular to radius)
2
Triangle OAP is right-angled, so use Pythagoras: PA² = OP² − OA² = 10² − 6²
AnswerPA = 8cm
PA and PB are both tangents from P, so PA = PB — triangle PAB is isosceles.
7. Alternate segment theorem
The angle between a tangent and a chord, drawn from the point of contact, equals the angle in the alternate segment (the angle subtended by the same chord from the other side of the circle).
The angle between the tangent and chord AB at A equals angle ACB, the angle in the alternate segment.
Practice questions
Work through each question before checking the answers. Where a question shows a circle diagram, the diagram is required to answer it.
Higher (Grade 5–7)
Q1AB is a diameter. Angle BAC = 35°. Find angle ABC.Higher
Diagram for Question 1 (not to scale).
Q2State the size of the angle in a semicircle.Higher
Q3Angle AOB at the centre of a circle is 100°. Find angle ACB at the circumference (C on the major arc).Higher
Diagram for Question 3 (not to scale).
Q4Angle ACB at the circumference is 42°. Find the angle AOB at the centre.Higher
Q5C and D are points on the same arc of a circle, with A and B fixed on the circle. Angle ADB = 38°. Find angle ACB.Higher
Higher — Hard (Grade 7–8)
Q6ABCD is a cyclic quadrilateral. Angle A = 110°. Find angle C.Grade 7–8
Q7ABCD is a cyclic quadrilateral. Angle A = 105°. Find angle C.Grade 7–8
Diagram for Question 7 (not to scale).
Q8O is the centre of a circle with radius 5cm. P is external with OP = 13cm. A tangent from P touches the circle at A. Find the length PA.Grade 7–8
Q9Two tangents from external point P touch a circle at A and B. Angle APB = 50°. Find angle PAB.Grade 7–8
Diagram for Question 9 (not to scale).
Q10A tangent meets a chord at the point of contact, making an angle of 48° with the chord. Find the angle in the alternate segment.Grade 7–8
Grade 9
Q11ABCD is a cyclic quadrilateral. Angle DAB = (3x + 10)° and angle DCB = (2x + 30)°. Find x, then find the size of angle DAB.Grade 9
Q12O is the centre of a circle with radius 8cm. P is external with OP = 17cm. A tangent from P touches the circle at A. Find the length PA.Grade 9
Q13A tangent meets a chord at the point of contact, making an angle of 42° with the chord. Find the angle in the alternate segment.Grade 9
Diagram for Question 13 (not to scale).
Q14O is the centre of a circle. Angle AOB (at the centre) = 130°. C is a point on the major arc. Find angle ACB, and state which theorem you used.Grade 9
Q15Two tangents from external point P touch a circle with centre O at A and B. Angle AOB = 140°. Find angle APB.Grade 9
Answers
Higher (Q1–Q5)
Q155°(angle ACB = 90°, so 180 − 90 − 35)
Q290°
Q350°(100 ÷ 2)
Q484°(42 × 2)
Q538°(angles in the same segment are equal)
Higher — Hard (Q6–Q10)
Q670°(180 − 110)
Q775°(180 − 105)
Q812cm(√(13² − 5²) = √144)
Q965°(triangle PAB is isosceles since PA = PB, so (180 − 50) ÷ 2)
Q1048°
Grade 9 (Q11–Q15)
Q11x = 28; angle DAB = 94°((3x+10)+(2x+30) = 180)
Q1215cm(√(17² − 8²) = √225)
Q1342°
Q1465°(angle at centre is twice angle at circumference: 130 ÷ 2)
Q1540°(quadrilateral OAPB: 360 − 90 − 90 − 140)
Common mistakes
Common Mistake 1
Applying the angle-in-a-semicircle rule without a diameter
The 90° rule only applies when the chord in question is a diameter — check the question states this (or that the line passes through the centre) before using it.
Common Mistake 2
Halving instead of doubling (or vice versa)
The angle at the centre is always the bigger one — double the angle at the circumference to get it, not the other way round. Sketch which angle is which before calculating.
Common Mistake 3
Forgetting cyclic quadrilateral angles must be opposite
Only opposite angles in a cyclic quadrilateral sum to 180° — adjacent angles do not follow this rule.
Common Mistake 4
Not stating which theorem was used
Circle theorem questions almost always award a mark for correctly naming the theorem used, not just for the correct number — always write the reason alongside your answer.
Exam tips
💡 Exam Tip 1
Mark on every angle you're given, and every one you find
As you work through a problem, write each angle directly onto the diagram — this makes it far easier to spot which theorem applies next.
💡 Exam Tip 2
Look for isosceles triangles formed by radii
Any triangle with two sides that are radii is isosceles (since all radii are equal) — this often unlocks an angle that isn't given directly by a named theorem.
💡 Exam Tip 3
Always give the reason, using the exact theorem name
Write reasons like "angle at the centre is twice the angle at the circumference" or "angles in the same segment are equal" — exact wording earns the reasoning mark.
💡 Exam Tip 4
Check for a right angle from a tangent before assuming you need trigonometry
Many "find the length" questions involving a tangent are really just Pythagoras in disguise — the tangent–radius right angle sets up the triangle for you.
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