Angle at the centre is twice the angle at the circumference
2xxcentre angle is double
Match the angles to the same chord or arc.
Add equal radii before calculating.
Write the theorem beside every angle you find.
Circle theorem visual atlas
Cover the rule, identify it from the picture, then say the full theorem aloud.
Centre and circumference
∠AOB=2∠ACB
Angle in a semicircle
∠ACB=90∘
Same segment
∠ACB=∠ADB
Cyclic quadrilateral
∠A+∠C=180∘
Radius and tangent
OT⊥ tangent
Alternate segment
∠BAT=∠ACB
Centre to a chord
OM⊥AB⇒AM=MB
Two tangents
PA=PB
Gold-standard guide
26 mins
What you will learn
Use angle rules inside circles.
Use a clear step-by-step method for circle theorems.
Check your answer and avoid the most common exam mistake.
Useful before you start
Core number skillsEarlier geometry and measures skillsShowing clear working
Core knowledge
Know the rule, then use it
These are the short notes. Read each one, then check you can use it in the worked example below.
Method
Method
Circle theorem questions test recognition before calculation
Step 1
Match the two angles to the same arc
Both angles stand on arc AB
Step 2
State the theorem before using it
The angle at the centre is twice the angle at the circumference on the same arc
Step 3
Form and solve the equation
116 = 2 × angle ACB, so angle ACB = 58 degrees
Watch out
Watch out
Do not choose a theorem because the picture looks familiar
f
Centre theorem
angleatcentre=2×angleatcircumference.
f
Cyclic quadrilateral
opposite angles sum to 180 degrees.
Worked example
O is the centre of a circle. A, B and C lie on the circumference. If angle AOB is 116 degrees, find angle ACB.
1
Match the two angles to the same arc: Both angles stand on arc AB.AOB is at the centre and ACB is at the circumference.
2
State the theorem before using it: The angle at the centre is twice the angle at the circumference on the same arc.
3
Form and solve the equation: 116 = 2 × angle ACB, so angle ACB = 58 degrees.
Final answer
Angle ACB = 58 degrees
Question ladder
Build up to the hardest questions
Do them in order. If you miss a step, read the solution, then redo the question without looking.
Workedreasoning
O is the centre of a circle. A, B and C lie on the circumference. If angle AOB is 116 degrees, find angle ACB.
3 marks4 minscircle-theorems-worked
Show solution
Worked solution
1.Match the two angles to the same arc: Both angles stand on arc AB.AOB is at the centre and ACB is at the circumference.
2.State the theorem before using it: The angle at the centre is twice the angle at the circumference on the same arc.
3.Form and solve the equation: 116 = 2 × angle ACB, so angle ACB = 58 degrees.
Final answer
Angle ACB = 58 degrees
Mark points
M1: match the two angles to the same arc
M1: state the theorem before using it
M1: form and solve the equation
A1: Angle ACB = 58 degrees
Watch out
Do not choose a theorem because the picture looks familiar.Trace both angle arms to the circumference and check that they meet the same two endpoints.A correct calculation with the wrong theorem will not earn the reasoning mark.
Diagnosticrecall
AB is a diameter and C lies on the circle. Find angle ACB.
1 mark2 minscircle-theorems-q1
Show solution
Worked solution
1.Spot the skill: Circle theorem questions test recognition before calculation.
2.Use the match the two angles to the same arc stage first, then state the theorem before using it.
3.Keep the final answer visible: 90 degrees, because the angle in a semicircle is 90 degrees..
Final answer
90 degrees, because the angle in a semicircle is 90 degrees.
Mark points
M1: use the correct circle theorem questions test recognition before calculation.first mark equal radii, diameters and tangents. then check which angles stand on the same chord or arc.the core rules are: the centre angle is twice the circumference angle; an angle in a semicircle is 90 degrees; angles in the same segment are equal; opposite angles in a cyclic quadrilateral total 180 degrees; a radius meets a tangent at 90 degrees; the alternate segment angle equals the angle in the opposite arc; a perpendicular from the centre bisects a chord; and two tangents from one point have equal lengths.
A1: 90 degrees, because the angle in a semicircle is 90 degrees.
Watch out
Do not choose a theorem because the picture looks familiar.Trace both angle arms to the circumference and check that they meet the same two endpoints.A correct calculation with the wrong theorem will not earn the reasoning mark.
Easyprocedure
ABCD is cyclic. Angle A is 73 degrees. Find angle C.
2 marks3 minscircle-theorems-q2
Show solution
Worked solution
1.Spot the skill: Circle theorem questions test recognition before calculation.
2.Use the state the theorem before using it stage first, then form and solve the equation.
3.Keep the final answer visible: 107 degrees, because opposite angles in a cyclic quadrilateral total 180 degrees..
Final answer
107 degrees, because opposite angles in a cyclic quadrilateral total 180 degrees.
Mark points
M1: use the correct circle theorem questions test recognition before calculation.first mark equal radii, diameters and tangents. then check which angles stand on the same chord or arc.the core rules are: the centre angle is twice the circumference angle; an angle in a semicircle is 90 degrees; angles in the same segment are equal; opposite angles in a cyclic quadrilateral total 180 degrees; a radius meets a tangent at 90 degrees; the alternate segment angle equals the angle in the opposite arc; a perpendicular from the centre bisects a chord; and two tangents from one point have equal lengths.
A1: 107 degrees, because opposite angles in a cyclic quadrilateral total 180 degrees.
Watch out
Do not choose a theorem because the picture looks familiar.Trace both angle arms to the circumference and check that they meet the same two endpoints.A correct calculation with the wrong theorem will not earn the reasoning mark.
Mediumreasoning
A tangent touches a circle at T. OT is a radius. Find the angle between OT and the tangent.
3 marks4 minscircle-theorems-q3
Show solution
Worked solution
1.Spot the skill: Circle theorem questions test recognition before calculation.
2.Use the form and solve the equation stage first, then match the two angles to the same arc.
3.Keep the final answer visible: 90 degrees, because a radius is perpendicular to a tangent at the point of contact..
Final answer
90 degrees, because a radius is perpendicular to a tangent at the point of contact.
Mark points
M1: use the correct circle theorem questions test recognition before calculation.first mark equal radii, diameters and tangents. then check which angles stand on the same chord or arc.the core rules are: the centre angle is twice the circumference angle; an angle in a semicircle is 90 degrees; angles in the same segment are equal; opposite angles in a cyclic quadrilateral total 180 degrees; a radius meets a tangent at 90 degrees; the alternate segment angle equals the angle in the opposite arc; a perpendicular from the centre bisects a chord; and two tangents from one point have equal lengths.
A1: 90 degrees, because a radius is perpendicular to a tangent at the point of contact.
Watch out
Do not choose a theorem because the picture looks familiar.Trace both angle arms to the circumference and check that they meet the same two endpoints.A correct calculation with the wrong theorem will not earn the reasoning mark.
Hardproblem solving
Angles APB and AQB stand on the same chord AB. If angle APB is 42 degrees, find angle AQB.
3 marks5 minscircle-theorems-q4
Show solution
Worked solution
1.Spot the skill: Circle theorem questions test recognition before calculation.
2.Use the match the two angles to the same arc stage first, then state the theorem before using it.
3.Keep the final answer visible: 42 degrees, because angles in the same segment are equal..
Final answer
42 degrees, because angles in the same segment are equal.
Mark points
M1: use the correct circle theorem questions test recognition before calculation.first mark equal radii, diameters and tangents. then check which angles stand on the same chord or arc.the core rules are: the centre angle is twice the circumference angle; an angle in a semicircle is 90 degrees; angles in the same segment are equal; opposite angles in a cyclic quadrilateral total 180 degrees; a radius meets a tangent at 90 degrees; the alternate segment angle equals the angle in the opposite arc; a perpendicular from the centre bisects a chord; and two tangents from one point have equal lengths.
A1: 42 degrees, because angles in the same segment are equal.
Watch out
Do not choose a theorem because the picture looks familiar.Trace both angle arms to the circumference and check that they meet the same two endpoints.A correct calculation with the wrong theorem will not earn the reasoning mark.
Exam-stylemulti-step
A tangent at A makes an angle of 64 degrees with chord AB. C lies in the opposite arc. Find angle ACB.
4 marks6 minscircle-theorems-q5
Show solution
Worked solution
1.Spot the skill: Circle theorem questions test recognition before calculation.
2.Use the state the theorem before using it stage first, then form and solve the equation.
3.Keep the final answer visible: 64 degrees, by the alternate segment theorem..
Final answer
64 degrees, by the alternate segment theorem.
Mark points
M1: use the correct circle theorem questions test recognition before calculation.first mark equal radii, diameters and tangents. then check which angles stand on the same chord or arc.the core rules are: the centre angle is twice the circumference angle; an angle in a semicircle is 90 degrees; angles in the same segment are equal; opposite angles in a cyclic quadrilateral total 180 degrees; a radius meets a tangent at 90 degrees; the alternate segment angle equals the angle in the opposite arc; a perpendicular from the centre bisects a chord; and two tangents from one point have equal lengths.
A1: 64 degrees, by the alternate segment theorem.
Watch out
Do not choose a theorem because the picture looks familiar.Trace both angle arms to the circumference and check that they meet the same two endpoints.A correct calculation with the wrong theorem will not earn the reasoning mark.
Exam-stylemulti-step
P is outside a circle. PA and PB are tangents. If PA = 7.5 cm, find PB.
4 marks6 minscircle-theorems-q6
Show solution
Worked solution
1.Spot the skill: Circle theorem questions test recognition before calculation.
2.Use the form and solve the equation stage first, then match the two angles to the same arc.
3.Keep the final answer visible: 7.5 cm, because tangents from the same external point are equal..
Final answer
7.5 cm, because tangents from the same external point are equal.
Mark points
M1: use the correct circle theorem questions test recognition before calculation.first mark equal radii, diameters and tangents. then check which angles stand on the same chord or arc.the core rules are: the centre angle is twice the circumference angle; an angle in a semicircle is 90 degrees; angles in the same segment are equal; opposite angles in a cyclic quadrilateral total 180 degrees; a radius meets a tangent at 90 degrees; the alternate segment angle equals the angle in the opposite arc; a perpendicular from the centre bisects a chord; and two tangents from one point have equal lengths.
A1: 7.5 cm, because tangents from the same external point are equal.
Watch out
Do not choose a theorem because the picture looks familiar.Trace both angle arms to the circumference and check that they meet the same two endpoints.A correct calculation with the wrong theorem will not earn the reasoning mark.
Exam-stylemulti-step
O is the centre. OM is perpendicular to chord AB and AB = 18 cm. Find AM.
4 marks6 minscircle-theorems-q7
Show solution
Worked solution
1.Spot the skill: Circle theorem questions test recognition before calculation.
2.Use the match the two angles to the same arc stage first, then state the theorem before using it.
3.Keep the final answer visible: 9 cm, because the perpendicular from the centre bisects the chord..
Final answer
9 cm, because the perpendicular from the centre bisects the chord.
Mark points
M1: use the correct circle theorem questions test recognition before calculation.first mark equal radii, diameters and tangents. then check which angles stand on the same chord or arc.the core rules are: the centre angle is twice the circumference angle; an angle in a semicircle is 90 degrees; angles in the same segment are equal; opposite angles in a cyclic quadrilateral total 180 degrees; a radius meets a tangent at 90 degrees; the alternate segment angle equals the angle in the opposite arc; a perpendicular from the centre bisects a chord; and two tangents from one point have equal lengths.
A1: 9 cm, because the perpendicular from the centre bisects the chord.
Watch out
Do not choose a theorem because the picture looks familiar.Trace both angle arms to the circumference and check that they meet the same two endpoints.A correct calculation with the wrong theorem will not earn the reasoning mark.
Exam-stylemulti-step
The angle at the centre is (5x + 10) degrees and the angle at the circumference on the same arc is (2x + 8) degrees. Find x.
4 marks6 minscircle-theorems-q8
Show solution
Worked solution
1.Spot the skill: Circle theorem questions test recognition before calculation.
2.Use the state the theorem before using it stage first, then form and solve the equation.
3.Keep the final answer visible: 5x + 10 = 2(2x + 8), so x = 6..
Final answer
5x + 10 = 2(2x + 8), so x = 6.
Mark points
M1: use the correct circle theorem questions test recognition before calculation.first mark equal radii, diameters and tangents. then check which angles stand on the same chord or arc.the core rules are: the centre angle is twice the circumference angle; an angle in a semicircle is 90 degrees; angles in the same segment are equal; opposite angles in a cyclic quadrilateral total 180 degrees; a radius meets a tangent at 90 degrees; the alternate segment angle equals the angle in the opposite arc; a perpendicular from the centre bisects a chord; and two tangents from one point have equal lengths.
A1: 5x + 10 = 2(2x + 8), so x = 6.
Watch out
Do not choose a theorem because the picture looks familiar.Trace both angle arms to the circumference and check that they meet the same two endpoints.A correct calculation with the wrong theorem will not earn the reasoning mark.
Grade 9 stretchproblem solving
AB is a diameter of a circle with centre O. C is on the circle and angle BAC = 34 degrees. A tangent is drawn at B. Find angle BOC and the acute angle between the tangent and BC. Give a reason for each answer.
3x+105x−6opposite angles sum to 180∘
4 marks7 minscircle-g9
Show solution
Worked solution
1.Use the angle-at-the-centre theorem on arc BC.
2.Use the fact that OB and OC are radii to identify an isosceles triangle.
3.Use the radius-tangent angle of 90 degrees, or the alternate segment theorem, for the second angle.
Final answer
angle BOC = 68 degrees and the acute angle between the tangent and BC = 34 degrees
A1: use 90 - 56 to obtain 34 degrees with a valid reason
Watch out
Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.
Timed checkpoint
16 mins - 9 marks
Switch between skills
Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.
1Circle theorems - 2 marksAB is a diameter and C lies on the circle. Find angle ACB.Mark answer
Answer
90 degrees, because the angle in a semicircle is 90 degrees.
2Angles, lines and polygons - 2 marksThe exterior angle of a regular polygon is 24°. How many sides?Mark answer
Answer
15
3Properties of shapes - 2 marksName all 2D shapes with equal diagonals that bisect each other.Mark answer
Answer
Rectangle, square
4Perimeter, area and volume - 3 marksFind the perimeter of a rectangle with length 13 cm and width 8 cm.Mark answer
Answer
42 cm
Mastery check
I can explain the method for circle theorems.
I can show clear working without skipping key steps.
I can avoid this mistake: Do not choose a theorem because the picture looks familiar.Trace both angle arms to the circumference and check that they meet the same two endpoints.A correct calculation with the wrong theorem will not earn the reasoning mark.
This guide follows the OCR GCSE Mathematics J560 specification. Practice questions are original Learnova questions shaped around official content and exam skills.