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Circles

Circles

Circle theorems

Question Rules Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. The four theorems
  4. Ways they test it
  5. Mistakes to avoid
  6. Summary
  7. Practice
  8. Can I do this?
  9. Quiz

In this note

Syllabus 4.7: circle theorems
  • 4.7The angle at the centre is twice the angle at the circumference.
  • 4.7Angles in the same segment are equal; opposite angles of a cyclic quadrilateral add up to 180°.
  • 4.7The alternate segment theorem - and give each theorem by name as a reason.

Before you start: Parts of a circle and the right-angle theorems. Next: Chords and tangents.

Formula bank

CentreLearn itthe angle at the centre is twice the angle at the circumferenceboth standing on the same arc.Use it when an angle at O and one on the circle from the same two points.
Same segmentLearn itangles in the same segment are equalsame chord, same side.Use it when a 'bow-tie' shape.
Cyclic quadrilateralLearn itopposite angles of a cyclic quadrilateral add up to 180°all four corners on the circle.Use it when a quadrilateral inside a circle.
Alternate segmentLearn itthe angle between a tangent and a chord equals the angle in the alternate segmentUse it when a tangent and a chord from the touching point.

The four theorems

Arrowhead

Angle at the centre is twice the angle at the circumferenceO2xxABC
Both angles stand on arc AB.

38° at C gives 76° at O.

Bow-tie

Angles in the same segment are equalyyABCD
C and D on the same side of chord AB.

Angle ACB = angle ADB.

Four corners on the circle

Opposite angles of a cyclic quadrilateral add up to 180°a180° − aABCD
ABCD is a cyclic quadrilateral.

Angle A = 72° → angle C = 108°.

Tangent + chord

Alternate segment theoremyyABCT
TA is a tangent at A.

Angle TAB = angle ACB.

General methodHow to answer a circle-theorem question

  1. Mark everything you know: the centre, radii (isosceles triangles), tangents (90° to radii), diameters.
  2. Pick a chord and look at the angles that stand on it: at the centre (× 2), on the same side (equal), on the opposite side (cyclic: 180°).
  3. A tangent with a chord from its touching point: alternate segment theorem.
  4. Write each value with its theorem name; finish with angles in a triangle or on a line.

Example A, B and C are on a circle centre O. Angle ABC = 52°. Find angle AOC and angle OAC.

Centre theorem, then isosceles OAC.

  1. angle AOC = 104° (angle at the centre is twice the angle at the circumference)
  2. OA = OC, so angle OAC = (180 − 104) ÷ 2

Answer 104° and 38°

Try it yourself

1
Angle at the circumference 47°. Angle at the centre on the same arc?
Show answer

Answer 94

2
Cyclic quadrilateral: one angle 113°. The opposite angle?
Show answer

Answer 67

3
Angle at the centre 140°. Angle at the circumference on the major arc?
Show answer

Answer 70

Ways they test it

Type 1 Find angles with reasons

"Find angle x. Give a geometrical reason for your answer." (2 marks per angle)

Spot it points on a circle, O marked.

  1. Name the theorem in full.
  2. 'Circle theorem' alone scores nothing.

Type 2 Cyclic quadrilateral with algebra

"Angle ABC = \((2x+10)^\circ\), angle ADC = \((3x-5)^\circ\). Find x." (3 marks)

Spot it opposite angles in x.

  1. Sum = 180.
  2. Solve.

For example \(5x+5=180\)

    Answer x = 35

    Type 3 Tangent and chord

    "TA is a tangent. Angle TAB = 64°, angle BAC = 50°. Find angle ABC." (3 marks)

    Spot it a tangent line through a point on the circle.

    1. Alternate segment: angle ACB = 64°.
    2. Angles in a triangle.

    For example angle ACB = 64°

      Answer angle ABC = 66°

      Mistakes to avoid

      • ✗ Halving to get the centre angle✓ centre = doubleThe angle at the centre is the bigger one.
      • ✗ Cyclic quad with O as a corner✓ all four corners on the circleO is not on the circle.
      • ✗ Same segment across the chord✓ same side onlyOpposite sides add to 180°.

      Summary

      • Centre = 2 × circumference (same arc).
      • Same segment: equal.
      • Cyclic quadrilateral: opposite angles add to 180°.
      • Tangent-chord angle = angle in the alternate segment.

      If you see …

      • O marked with lines to the circlethink centre theorem and isosceles triangles.
      • four points on the circle joined upcyclic quadrilateral.

      Practice

      Practice: easy, medium, hard

      Easy

      1
      Angle at the circumference 38°. Angle at the centre on the same arc?
      Show answer

      Answer 76

      2
      Two angles stand on the same chord, same side. One is 47°. The other?
      Show answer

      Answer 47

      3
      Cyclic quadrilateral: angle A = 72°. Angle C?
      Show answer

      Answer 108

      Medium

      1
      A, B, C on a circle centre O. Angle ABC = 52°. Find angle OAC.
      Guide me

      What is it asking? an angle

      What do we know? angle AOC = 104°

      Plan isosceles OAC

      Show working
      1. (180 − 104) ÷ 2 = 38°

      Answer 38

      2
      Cyclic quadrilateral ABCD: angle ABC = \((2x+10)^\circ\), angle ADC = \((3x-5)^\circ\). Find angle ADC.
      Guide me

      What is it asking? an angle

      What do we know? opposite: sum 180

      Plan \(5x+5=180\Rightarrow x=35\)

      Show working
      1. 3(35) − 5 = 100°

      Answer 100

      3
      TA is a tangent at A. Angle TAB = 64°. Angle ACB (C on the circle, other side of AB)?
      Guide me

      What is it asking? an angle

      What do we know? alternate segment theorem

      Plan equal

      Show working
      1. 64°

      Answer 64

      Hard

      1
      TA is a tangent at A; angle TAB = 64°, angle BAC = 50°. Find angle ABC.
      Guide me

      What is it asking? an angle

      What do we know? angle ACB = 64° (alternate segment)

      Plan 180 − 64 − 50

      Show working
      1. 66°

      Answer 66

      PartAnswer / workingMarksGuidance
      angle ACB = 64° with reasonB1
      180 − 64 − 50M1
      66A1
      Total: 3 marks
      2
      A, B, C, D on a circle centre O. Angle BOD = 136°, A on the major arc. Find angle BAD.
      Guide me

      What is it asking? an angle

      What do we know? centre theorem

      Plan 136 ÷ 2

      Show working
      1. 68°

      Answer 68

      PartAnswer / workingMarksGuidance
      68 with reasonB2
      Total: 2 marks
      3
      With C on the minor arc of the same circle, find angle BCD.
      Guide me

      What is it asking? an angle

      What do we know? ABCD is cyclic

      Plan 180 − 68

      Show working
      1. 112°

      Answer 112

      PartAnswer / workingMarksGuidance
      112 with reasonB2
      Total: 2 marks

      Can I do this?

      Anything unticked: revise that section first. All ticked: take the quiz.

      Quiz

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