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Maths · Circle Theorems

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Angles at the Centre and in a Semicircle

Using the angle at the centre and the angle in a semicircle, with reasons and radii.

  • Higher
  • 9 key terms
  • All boards

Learning Objectives

  1. 1Name the parts of a circle: centre, radius, diameter, chord, arc, segment, sector and tangent.
  2. 2Use the theorem that the angle at the centre is twice the angle at the circumference.
  3. 3Use the theorem that the angle in a semicircle is a right angle.
  4. 4Give a reason for every angle, using the exact words of the theorem.

Angles in circles

Circle theorems are rules about the angles made by lines drawn inside a circle. This is Higher tier content on every board, and the questions are always the same kind: a diagram with some angles given, and a request to find another angle and give a reason. The marks are for the angle and for the reason, so you must learn the theorems in their exact wording and write each reason in a short sentence. This first lesson covers the two theorems that involve the centre, and the later lessons add the others.

Parts of a circle

The words are used in every circle theorem question.

  • Centre, radius and diameter

    The radius goes from the centre to the edge, and the diameter goes through the centre from edge to edge.

  • Chord

    A straight line joining two points on the circle. The diameter is the longest chord.

  • Arc

    A part of the circumference.

  • Tangent

    A straight line that touches the circle at exactly one point.

The angle at the centre

The angle at the centre is twice the angle at the circumference, when both angles are made by the same arc.

  • The theorem

    If \(A\), \(B\) and \(C\) are on a circle with centre \(O\), then angle \(AOB\) is twice angle \(ACB\).

  • Same arc

    Both angles are made from the chord \(AB\), with the angle at \(C\) on the other side from the arc \(AB\) that makes the angle at \(O\).

  • Using it

    If the angle at the centre is \(100^\circ\), the angle at the circumference is \(50^\circ\). If the angle at the circumference is \(35^\circ\), the angle at the centre is \(70^\circ\).

  • The reason

    "The angle at the centre is twice the angle at the circumference."

Using the centre theorem

\(A\), \(B\) and \(C\) are points on a circle with centre \(O\). Angle \(AOB = 112^\circ\). Work out angle \(ACB\), and give a reason.

Show the solutionHide the solution
  1. 1 Identify \(AOB\) is the angle at the centre, and \(ACB\) is the angle at the circumference, both on the chord \(AB\).
  2. 2 Halve \(112 \div 2 = 56\).
  3. 3 Write the answer Angle \(ACB = 56^\circ\).
  4. 4 Reason The angle at the centre is twice the angle at the circumference.

Answer\(56^\circ\), because the angle at the centre is twice the angle at the circumference

The angle in a semicircle

The angle in a semicircle is \(90^\circ\).

  • The theorem

    If \(AB\) is a diameter and \(C\) is any other point on the circle, then angle \(ACB = 90^\circ\).

  • Why

    The angle at the centre on a diameter is \(180^\circ\), and half of that is \(90^\circ\).

  • Using it

    Once you have a right-angled triangle, the other two angles add up to \(90^\circ\), and you can use Pythagoras too.

  • The reason

    "The angle in a semicircle is \(90^\circ\)."

Using the semicircle theorem

\(AB\) is a diameter of a circle. \(C\) is a point on the circle, and angle \(CAB = 34^\circ\). Work out angle \(ABC\), and give reasons.

Show the solutionHide the solution
  1. 1 Right angle Angle \(ACB = 90^\circ\), because the angle in a semicircle is \(90^\circ\).
  2. 2 Angles in a triangle \(34^\circ + 90^\circ = 124^\circ\).
  3. 3 Subtract \(180^\circ - 124^\circ = 56^\circ\).
  4. 4 Reasons The angle in a semicircle is \(90^\circ\), and the angles in a triangle add up to \(180^\circ\).

Answer\(56^\circ\)

Test yourself

  1. 1

    What is a chord?

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    A straight line joining two points on a circle.

  2. 2

    What is the angle at the centre in terms of the angle at the circumference?

    Show answerHide answer

    Twice as big.

  3. 3

    What is the angle in a semicircle?

    Show answerHide answer

    \(90^\circ\).

  4. 4

    What line makes the angle in a semicircle?

    Show answerHide answer

    A diameter.

  5. 5

    What do you give with every angle in the exam?

    Show answerHide answer

    A reason.

Exam technique: circle theorems

The reason is worth as much as the angle.

  • Use the exact wording

    "Angle at the centre is twice the angle at the circumference."

  • One reason per step

    Do not combine several reasons in one sentence.

  • Mark the diagram

    Write each angle you find on the diagram.

  • Look for isosceles triangles

    Two radii make an isosceles triangle, a useful fact for later lessons.

Summary and exam focus

  • The angle at the centre is twice the angle at the circumference on the same arc.
  • The angle in a semicircle is \(90^\circ\).
  • Every angle needs a reason, using the exact words.
  • Two radii form an isosceles triangle.

Exam focus

\(A\), \(B\) and \(C\) are points on a circle with centre \(O\). Angle \(ACB = 38^\circ\). Work out angle \(AOB\), and give a reason for your answer. (2 marks) (2 marks)

The angle at the centre is twice the angle at the circumference, so \(AOB = 2 \times 38 = 76^\circ\). Write the reason in full: "The angle at the centre is twice the angle at the circumference."

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Circle theorem
A rule about angles and lines in a circle.
Chord
A straight line joining two points on a circle.
Arc
A part of the circumference.
Tangent
A line that touches a circle at one point.
Diameter
A chord through the centre of a circle.
Circumference
The distance around a circle, or the circle itself.
Subtend
To make an angle from a chord or arc at a point.
Semicircle
Half of a circle, cut by a diameter.
Reason
A statement of the theorem that justifies an angle.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Calculate 2 marks Core

The diagram is not drawn to scale. \(A\), \(B\) and \(C\) are points on a circle, centre \(O\). Angle \(ACB = 36^\circ\). Calculate angle \(AOB\), giving a reason for your answer. [2 marks]

A circle diagram showing the angle at the centre and the angle at the circumference.

Mark scheme — 2 marks available

  • \(72\) — B1
  • The angle at the centre is twice the angle at the circumference — B1

Model answer

Angle \(AOB = 2 \times 36 = 72^\circ\), because the angle at the centre is twice the angle at the circumference.

2. Exam question Calculate 3 marks Core

The diagram is not drawn to scale. \(AB\) is a diameter of a circle and \(C\) is a point on the circle. Angle \(BAC = 27^\circ\). Calculate angle \(ABC\), giving a reason for each step. [3 marks]

A circle diagram showing a triangle in a semicircle.

Mark scheme — 3 marks available

  • Angle \(ACB = 90^\circ\) — B1
  • The angle in a semicircle is a right angle — B1
  • \(63\) — B1

Model answer

Angle \(ACB = 90^\circ\), because the angle in a semicircle is a right angle. Then \(ABC = 180 - 90 - 27 = 63^\circ\).

3. Exam question Calculate 3 marks Core

\(A\), \(B\) and \(C\) are points on a circle, centre \(O\), with \(C\) on the major arc \(AB\). Angle \(AOB = 7x - 6\) and angle \(ACB = 3x + 4\). Calculate the value of \(x\). [3 marks]

Mark scheme — 3 marks available

  • \(7x - 6 = 2(3x + 4)\) — M1
  • \(7x - 6 = 6x + 8\) — M1
  • \(14\) — A1

Model answer

The angle at the centre is twice the angle at the circumference, so \(7x - 6 = 2(3x + 4) = 6x + 8\). Then \(x = 14\).

4. Exam question Calculate 4 marks Core

\(A\), \(B\) and \(C\) are points on a circle, centre \(O\), with \(C\) on the major arc \(AB\). Angle \(OAB = 22^\circ\). Calculate angle \(ACB\), giving reasons for your answer. [4 marks]

Mark scheme — 4 marks available

  • \(180 - 22 - 22\) or angle \(OBA = 22^\circ\) — M1
  • \(136\) — A1
  • \(68\) — A1
  • Radii make an isosceles triangle, and the angle at the centre is twice the angle at the circumference — B1

Model answer

\(OA = OB\) because they are radii, so angle \(OBA = 22^\circ\). Angle \(AOB = 180 - 22 - 22 = 136^\circ\). The angle at the circumference is half the angle at the centre, so \(ACB = 68^\circ\).

5. Exam question Calculate 4 marks Stretch

\(AB\) is a diameter of a circle and \(C\) is a point on the circle. \(AC = 5\) cm and \(BC = 12\) cm. Calculate the radius of the circle, giving a reason for your answer. [4 marks]

Mark scheme — 4 marks available

  • The angle in a semicircle is a right angle — B1
  • \(5^2 + 12^2\) or \(25 + 144\) — M1
  • \(AB = 13\) — A1
  • \(6.5\) — A1

Model answer

Angle \(ACB = 90^\circ\), because the angle in a semicircle is a right angle. \(AB^2 = 5^2 + 12^2 = 25 + 144 = 169\), so \(AB = 13\) cm and the radius is \(6.5\) cm.

6. Exam question Calculate 3 marks Stretch

The diagram is not drawn to scale. \(A\), \(B\) and \(C\) are points on a circle, centre \(O\). Angle \(AOB = 96^\circ\), and \(C\) is on the minor arc \(AB\). Calculate angle \(ACB\), giving a reason for your answer. [3 marks]

A circle diagram showing the angle at the circumference on the minor arc.

Mark scheme — 3 marks available

  • \(360 - 96 = 264\) — M1
  • \(132\) — A1
  • The angle at the centre is twice the angle at the circumference — B1

Model answer

The reflex angle \(AOB = 360 - 96 = 264^\circ\). The angle at the circumference is half of this, so \(ACB = 132^\circ\).

7. Multiple choice 1 mark Easier

The angle at the centre of a circle is \(84^\circ\). What is the angle at the circumference made by the same arc?

  1. A \(168^\circ\)
  2. B \(42^\circ\) Correct
  3. C \(84^\circ\)
  4. D \(96^\circ\)

Why: The angle at the centre is twice the angle at the circumference, so the angle at the circumference is \(84 \div 2 = 42^\circ\).

8. Multiple choice 1 mark Easier

What is the size of an angle in a semicircle?

  1. A \(90^\circ\) Correct
  2. B \(180^\circ\)
  3. C \(45^\circ\)
  4. D \(60^\circ\)

Why: The angle at the centre on a diameter is \(180^\circ\), so the angle at the circumference is half of it, \(90^\circ\).

9. Multiple choice 1 mark Easier

Which reason fits this statement? \(\angle AOB = 2 \times \angle ACB\)

  1. A Angles in the same segment are equal
  2. B The angle in a semicircle is \(90^\circ\)
  3. C Opposite angles of a cyclic quadrilateral add up to \(180^\circ\)
  4. D The angle at the centre is twice the angle at the circumference Correct

Why: The angle at \(O\), the centre, is double the angle at \(C\), on the circumference.

10. Multiple choice 1 mark Core

\(AB\) is a diameter and \(C\) is on the circle. Angle \(CAB = 35^\circ\). What is angle \(CBA\)?

  1. A \(35^\circ\)
  2. B \(90^\circ\)
  3. C \(55^\circ\) Correct
  4. D \(145^\circ\)

Why: Angle \(ACB = 90^\circ\) in a semicircle, so \(CBA = 180 - 90 - 35 = 55^\circ\).

11. Multiple choice 1 mark Core

The angle at the circumference is \(3x\) and the angle at the centre on the same arc is \(5x + 20\). What is \(x\)?

  1. A \(10\)
  2. B \(20\) Correct
  3. C \(4\)
  4. D \(40\)

Why: \(5x + 20 = 2 \times 3x = 6x\), so \(x = 20\).

12. Multiple choice 1 mark Core

In triangle \(OAB\), \(O\) is the centre and \(\angle OAB = 28^\circ\). What is angle \(AOB\)?

  1. A \(124^\circ\) Correct
  2. B \(62^\circ\)
  3. C \(56^\circ\)
  4. D \(152^\circ\)

Why: \(OA = OB\) are radii, so \(\angle OBA = 28^\circ\) and \(\angle AOB = 180 - 28 - 28 = 124^\circ\).

13. Multiple choice 1 mark Core

\(AB\) is a diameter of length 10 cm and \(AC = 6\) cm. What is \(BC\)?

  1. A 4 cm
  2. B \(\sqrt{136}\) cm
  3. C 16 cm
  4. D 8 cm Correct

Why: Angle \(ACB = 90^\circ\) in a semicircle, so \(BC^2 = 10^2 - 6^2 = 64\) and \(BC = 8\).

14. Multiple choice 1 mark Stretch

Two points \(A\) and \(B\) are on a circle with centre \(O\), and \(\angle AOB = 130^\circ\). \(C\) is on the minor arc \(AB\). What is angle \(ACB\)?

  1. A \(65^\circ\)
  2. B \(130^\circ\)
  3. C \(115^\circ\) Correct
  4. D \(230^\circ\)

Why: The reflex angle at the centre is \(360 - 130 = 230^\circ\), so \(ACB = 230 \div 2 = 115^\circ\).

15. Multiple choice 1 mark Stretch

Which statement is true for every triangle with a diameter as one side and its third corner on the circle?

  1. A It is isosceles
  2. B It is right-angled Correct
  3. C It is equilateral
  4. D It has an angle of \(60^\circ\)

Why: The angle opposite the diameter is in a semicircle, so it is always \(90^\circ\).