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Circle Theorem Angle at the Centre

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Written Lesson

A circle theorem describes a relationship between angles, chords and arcs. This lesson examines the relationship between an angle formed at the centre and an angle formed at the circumference.

What You Will Learn

  • Identify angles at the centre and at the circumference.
  • Recognise when two angles stand on the same arc.
  • State and apply the angle-at-the-centre theorem.
  • Calculate missing angles and unknown algebraic values.
  • Recognise when a reflex central angle is required.
  • Give the correct geometric reason.

Key Vocabulary

TermMeaning
CentreThe fixed point in the middle of a circle.
CircumferenceThe boundary of a circle.
RadiusA line segment joining the centre to the circumference.
ChordA line segment joining two points on the circumference.
ArcPart of the circumference between two points.
Minor arcThe shorter arc between two points.
Major arcThe longer arc between two points.
Angle at the centreAn angle whose vertex is the centre and whose arms are radii.
Angle at the circumferenceAn angle whose vertex is on the circumference and whose arms are chords.
SubtendTo form an angle by joining the endpoints of an arc or chord to a point.
Same arcTwo angles with arms that reach the same two endpoints on the circle.
Reflex central angleThe central angle greater than 180° and less than 360°.

Angles Subtended by an Arc

An arc or chord subtends an angle when lines are drawn from its endpoints to another point. In the diagram, ∠AOB and ∠ACB both have endpoints A and B, so they stand on the same arc AB.

ABOC2θθarc AB
Both angles stand on arc AB: the central angle is twice the circumference angle.
To identify the arc: trace the two arms of the angle to the circumference. The endpoints identify the arc or chord.

The Angle-at-the-Centre Theorem

The angle which an arc of a circle subtends at the centre is twice the angle it subtends at any point on the remaining part of the circumference.

In shorter form: the angle at the centre is twice the angle at the circumference when both angles stand on the same arc.

∠AOB = 2 × ∠ACB
∠ACB = ∠AOB2

Identifying the Two Angles

  • Angle at the centre: its vertex is at the centre O and its arms are radii.
  • Angle at the circumference: its vertex lies on the circumference and its arms are chords.
In ∠ACB, the middle letter C is the vertex. The endpoints are A and B.

The Same-Arc Requirement

The theorem applies only when the central angle and the circumference angle have the same two endpoints.

ABOCsame endpoints: A and B
Trace each pair of arms: both pairs end at A and B.

Worked Examples

Example 1: Finding the Angle at the Centre

The angle at the circumference is 34°. Find the central angle standing on the same arc.

ABOCx°34°
The geometry is drawn to show the exact 34° and 68° relationship.

∠AOB = 2 × 34° = 68°

Answer: 68°

Example 2: Finding the Angle at the Circumference

The central angle is 126°. Find the circumference angle standing on the same arc.

Angle at circumference = 126° ÷ 2 = 63°

Answer: 63°

Example 3: Working Backwards

The circumference angle is x°, and the central angle is 96°.

96 = 2x

x = 96 ÷ 2 = 48

Answer: x = 48°

Reflex Angles at the Centre

If the point at the circumference lies on the minor arc, the circumference angle stands on the major arc. The corresponding central angle is therefore the reflex angle.

ABOC220° reflex140°110°
Point C is on the minor arc, so ∠ACB intercepts the major arc and corresponds to the 220° reflex central angle.
An obtuse angle at the circumference produces a reflex angle at the centre.

Example 4: Finding Both Central Angles

The circumference angle is 110°. Find both central angles.

Reflex ∠AOB = 2 × 110° = 220°

Smaller ∠AOB = 360° − 220° = 140°

Answers: 220° reflex and 140° smaller angle.

Example 5: Choosing the Reflex Angle

The smaller central angle is 136°, and C lies on the minor arc AB. Find ∠ACB.

Because C lies on the minor arc, use the reflex central angle:

Reflex ∠AOB = 360° − 136° = 224°

∠ACB = 224° ÷ 2 = 112°

Answer: 112°

Writing a Complete Geometric Solution

∠AOB = 2 × ∠ACB = 2(37°) = 74°

Reason: The angle at the centre is twice the angle at the circumference standing on the same arc.

The Angle-at-the-Centre Method

  1. Locate the centre O.
  2. Identify the central angle.
  3. Identify the circumference angle.
  4. Check that both angles have the same endpoints.
  5. Decide whether the smaller or reflex central angle is required.
  6. Double the circumference angle or halve the central angle.
  7. State the theorem as the reason.

Common Mistakes

  • Doubling the central angle instead of the circumference angle.
  • Using angles that do not stand on the same arc.
  • Ignoring the middle letter when identifying a vertex.
  • Using the smaller central angle when the reflex angle is required.
  • Forgetting to halve the central angle.
  • Failing to state the theorem as the reason.
  • Assuming that any point inside the circle is the centre.
  • Trusting a diagram that is not drawn to scale.

Check for Understanding

1. An angle at the circumference is 41°. What is the central angle standing on the same arc?

  1. 20.5°
  2. 82°
  3. 41°
  4. 123°

Answer: B. 82°.

Explanation: Double the circumference angle: 2 × 41° = 82°.

2. A central angle is 138°. What is the angle at the circumference on the same arc?

  1. 276°
  2. 46°
  3. 78°
  4. 69°

Answer: D. 69°.

Explanation: Halve the central angle: 138° ÷ 2 = 69°.

3. When can the angle-at-the-centre theorem be applied?

  1. When both angles have the same endpoints and stand on the same arc
  2. Whenever both angles are inside the circle
  3. Only when both angles are acute
  4. Whenever one angle is twice another

Answer: A.

Explanation: The matching endpoints show that the two angles are subtended by the same arc or chord.

4. A central angle is 6x°, and the circumference angle on the same arc is 51°. What is x?

  1. 8.5
  2. 12
  3. 17
  4. 34

Answer: C. 17.

Explanation: 6x = 2(51) = 102, so x = 17.

5. The smaller central angle is 154°, and point C lies on the minor arc between the endpoints. What is the obtuse angle at C?

  1. 77°
  2. 103°
  3. 206°
  4. 257°

Answer: B. 103°.

Explanation: The reflex central angle is 360° − 154° = 206°; then 206° ÷ 2 = 103°.

Practice Exercises

  1. An angle at the circumference is 29°. Find the central angle on the same arc.
  2. A central angle is 114°. Find the circumference angle on the same arc.
  3. ∠ACB = 47°. Find ∠AOB when both stand on arc AB.
  4. ∠POQ = 152°. Find ∠PRQ on the same arc.
  5. The central angle is 4x°, and the circumference angle is 36°. Find x.
  6. The central angle is (3x + 8)°, and the circumference angle is (x + 14)°. Find x.
  7. An angle at the circumference standing on a major arc is 104°. Find the reflex central angle.
  8. A circumference angle is 115°. Find the reflex and smaller central angles.
  9. The smaller central angle is 128°. A point C lies on the minor arc. Find the obtuse angle at C.
  10. Explain why the theorem cannot be used when the two angles have different endpoints.

Practice Answers

  1. 58°
  2. 57°
  3. 94°
  4. 76°
  5. 4x = 72; x = 18
  6. 3x + 8 = 2(x + 14); x = 20
  7. 208°
  8. 230° reflex; 130° smaller
  9. 360° − 128° = 232°; 232° ÷ 2 = 116°
  10. The theorem applies only when both angles stand on the same arc or chord.

Lesson Summary

  • The central angle is twice the circumference angle standing on the same arc.
  • Both angles must have the same endpoints.
  • Multiply the circumference angle by 2 to find the central angle.
  • Divide the central angle by 2 to find the circumference angle.
  • An obtuse circumference angle corresponds to a reflex central angle.
  • Always state the circle theorem used.

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