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GCSE Maths Geometry and Measure

Circle Theorems

Angle at the Centre

Angle at the Centre Is Twice the Angle at the Circumference

Here we will learn about the circle theorem: the angle at the centre, including its application, proof, and using it to solve more difficult problems.

There are also circle theorem worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is the angle at the centre?

The angle at the centre is twice the angle at the circumference of a circle.

Angle at the centre is twice the angle at the circumference image 1 1

In the diagram above, the two points B and D on the circumference are joined to the centre, C , and to another point on the circumference, A . The angle BCD is twice the size of the angle BAD . Therefore, the angle at the centre is twice the angle at the circumference.

There are multiple ways of viewing this theorem as it depends on the location of the point A on the circumference. 

Below, we have moved point A on the circumference to show you how this theorem can appear in different circles.

Angle at the centre is twice the angle at the circumference image 2 1

What is the angle at the centre?

What is the angle at the centre?

Key parts of a circle needed for this theorem

Angle at the centre is twice the angle at the circumference image 3 1

  • The centre  of the circle is a point that locates the middle of a circle.
  • The circumference of the circle is the distance around the edge of the circle.
  • A chord is a straight line that meets the circumference in two places. The longest chord in a circle is the diameter.

Proving that the angle at the centre is twice the angle at the circumference

To be able to prove this theorem, you do not need to know any other circle theorem. You just need to be confident with angles in a triangle and angles around a point.

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Diagram
 
Angle at the centre is twice the angle at the circumference image 4 1
 
Angle at the centre is twice the angle at the circumference image 4 2
 
Angle at the centre is twice the angle at the circumference image 4 3
 
Angle at the centre is twice the angle at the circumference image 4 4
 
Angle at the centre is twice the angle at the circumference image 4 5
 
Angle at the centre is twice the angle at the circumference image 4 6
 
 
 
Angle at the centre is twice the angle at the circumference image 4 7
 
Angle at the centre is twice the angle at the circumference image 4 8
 
Angle at the centre is twice the angle at the circumference image 4 9
 
Angle at the centre is twice the angle at the circumference image 4 10
 
 
 
Angle at the centre is twice the angle at the circumference image 4 11
 
 
 
 
 
Angle at the centre is twice the angle at the circumference image 4 12
 
 
 

      Description
 
      A, B, and D are points on the
      circumference and C is the centre
      of the circle.
 
      BC and CD are radii.
 
      AB and AD are chords.
 
 
 
      Connect the points AC so that you have
      two triangles (ACD and ABC).
 
 
 
 
 
 
 
      Let’s inspect triangle ACD.
 
 
 
 
 
 
 
 
      We know that AC and CD are radii and
      so this is an isosceles triangle. We can
      therefore state the angles at A and D
      are equal, so we have labelled them x.
 
 
 
 
      As angles in a triangle add up to 180^o ,
      the angle at C will be 180 degrees
      minus the sum of the other two angles.
      This is expressed as 180-2x.
 
 
 
 
 
      Now let us inspect triangle ABC.
 
 
 
 
 
 
 
 
      BC and AC are radii and so we have
      another isosceles triangle. As we
      cannot say that the angles at A and B
      are the same as angle x in triangle
      ACD, we say that both the angles are
      equal to another number, y.
 
 
 
      As angles in a triangle total 180^o , we
      can state that the angle at C is
       180-2y for the same reason as in
       step 5.
 
 
 
 
 
      Now we have the diagram with all the
      angles filled in. We still need to prove
      that the external angle to BCD is equal
      to twice the angle at A but for now, let
      us simplify each angle within ABCD.
 
 
 
      The angle at A is now x+y and the
      angle at C is the sum of 180-2x and
       180-2y (or 180-2x+180-2y ).
      As we need to know the angle on the
      other side of BCD, we can use the fact
      that angles around a point total 360^o.
 
 
 
       \begin{aligned} &360-(360-2x-2y) \\ &=360-360+2x+2y \\ &=2x+2y \\ \end{aligned}
 
      We now have the external angle BCD
      equalling 2x+2y.
 
      If we factorise this expression, we get
      our angle BCD as 2(x+y)
 
 
 
 
      So as angle BAD =x+y and angle
      BCD =2(x+y) , the angle at the centre
      is twice the angle at the circumference.

How to use the angle at the centre theorem

In order to use the fact that the angle at the centre is twice the angle at the circumference

  1. Locate the key parts of the circle for the theorem
  2. Use other angle facts to determine the angle at the centre or the angle at the circumference
  3. Use the angle at the centre theorem to state the other missing angle

Explain how to use the angle at the centre theorem

Explain how to use the angle at the centre theorem

Angle at the centre worksheet

Get your free angle at the centre worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOON
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Angle at the centre worksheet

Get your free angle at the centre worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOON

Angle at the centre examples

Example 1: standard diagram

ABCD is an arrowhead where C is the centre of the circle and A, B, and D lie on the circumference. Calculate the size of angle BAD .

Angle at the centre is twice the angle at the circumference example 1 1

  1. Locate the key parts of the circle for the theorem.

Angle at the centre is twice the angle at the circumference example 1 step 1 1

Here we have:

  • The angle BCD = 150^o
  • BC = CD = Radii
  • AB and AD are chords
  • The angle BAD = \theta

2Use other angle facts to determine the angle at the centre or the angle at the circumference.

We already know that BCD = 150^o so we do not need to use any other angle fact to determine this angle for this example.

3Use the angle at the centre theorem to state the other missing angle.

The angle at the centre is twice the angle at the circumference and so as we know the angle at the centre, we need to divide this number by 2 to get the angle BAD :

BAD = 150 ÷ 2

BAD = 75°

Example 2: angles in a triangle

Below is a circle with centre C . Points A, B, and D are on the circumference of the circle. Calculate the size of angle \theta.

Angle at the centre is twice the angle at the circumference example 2 1

Angle at the centre is twice the angle at the circumference example 2 step 1 1


Here we have:

  • AC = BC = CD = Radii
  • AB is a chord
  • AD is the diameter
  • The angle ABC = 46^o
  • The angle BCD = \theta

As AC = BC , we can say that ABC is an isosceles triangle. This means that the angle BAC = angle ABC = 46^o .


Angle at the centre is twice the angle at the circumference example 2 step 2 1

The angle at the centre is twice the angle at the circumference and so as we know the angle at the circumference BAD , we need to multiply this number by 2 to get the angle BCD at the centre:


BCD = 46 × 2

BCD = 92°

Example 3: cyclic quadrilateral

ABCD is a cyclic quadrilateral around the centre E . Calculate the size of the reflex angle AEC , labelled \theta.

Angle at the centre is twice the angle at the circumference example 3 1

Angle at the centre is twice the angle at the circumference example 3 step 1 1


Here we have:

  • AC = BC = CD = Radii
  • AD and CD are chords
  • The angle ADC = 72^o
  • The reflex angle AEC = \theta

As ABCD is a cyclic quadrilateral, opposite angles in a cyclic quadrilateral total 180^o . This means that angle ABC = 180 - 72 = 108^o . We therefore have the updated diagram:

Angle at the centre is twice the angle at the circumference example 3 step 2 1

The angle at the centre is twice the angle at the circumference and so as we know the angle at the circumference ABC , we need to multiply this number by 2 to get the angle AEC at the centre:

AEC = 108 × 2

AEC = 216°

Example 4: alternative case (angle at the centre)

A, B, and D are points on the circumference of the circle with centre C . Calculate the value of \theta.

Angle at the centre is twice the angle at the circumference example 4 1

Here we have:

  • BC = CD = Radii
  • AB, AD and BD are chords
  • The angle   BDA = 39^o
  • The angle ABD = 104^o
  • The angle BCD = \theta

If we can calculate the size of BAD , we can then use the angle at the centre theorem to calculate BCD . As ABD is a triangle, we can calculate the missing angle BAD as angles in a triangle total = 180^o .

BAD = 18 - (104 + 39)

BAD = 37°

Angle at the centre is twice the angle at the circumference example 4 step 2 1

The angle at the centre is twice the angle at the circumference and so as we know the angle at the circumference BAC , we need to multiply this number by 2 to get the angle BCD at the centre:


BCD = 37 × 2

BCD = 74°

Example 5: tangent of a circle

ABCD is a quadrilateral inscribed in a circle with centre C . EF and CG are tangents that touch the circle at B and D . The tangents intersect at the point P . Calculate the size of angle BAD .

Angle at the centre is twice the angle at the circumference example 5 1

Angle at the centre is twice the angle at the circumference example 5 step 1 1


Here we have:

  • BC = CD = Radii
  • AB and AD are chords
  • The angle FPG = 156^o
  • The angle BAD = \theta

If we can calculate the size of BCD , we can then use the angle at the centre theorem to calculate BAD . As EPF is a straight line, we can calculate the angle BPD and then use the tangent theorem to work out BCD . As angles on a straight line total = 180^o .

BPD = 180 - 156

BPD = 24°

Angle at the centre is twice the angle at the circumference example 5 step 2 1


The angle between the tangent and the radius is 90^o so angle CBP = angle CDP = 90^o


Angle at the centre is twice the angle at the circumference example 5 step 2 (2) 1


BCDP is a kite. Angles in a quadrilateral total 360^o so we can work out BCD :

BCD = 360 - (90 + 90 + 24)

BCD = 66°


We now have the following information added to the diagram:


Angle at the centre is twice the angle at the circumference example 5 step 2 (3) 1

The angle at the centre is twice the angle at the circumference and so as we know the angle at the circumference BCD , we need to multiply this number by 2 to get the angle BAD at the centre:


\begin{aligned} &BAD=66 \div 2 \\\\ &BAD=33^o \end{aligned}

Example 6: alternate segment theorem

ABC is a triangle inscribed in the circle with centre O . The tangent DE touches the circle at point C . Calculate the size of angle BOC .

Angle at the centre is twice the angle at the circumference example 6 1

Angle at the centre is twice the angle at the circumference example 6 step 1 1


Here we have:

  • OB = OC = Radii
  • AB, AC and BC are chords
  • The angle BCE = 82^o
  • The angle BOC = \theta

The angle BAC is in the alternate segment to the angle BCE and so as angles in the alternate segment are equal, angle BAC = 82^o .


Angle at the centre is twice the angle at the circumference example 6 step 2 1

The angle at the centre is twice the angle at the circumference and so as we know the angle at the circumference BAC , we need to multiply this number by 2 to get the angle BOD at the centre:

BOC = 82 × 2

BOC = 164 °


Common misconceptions

  • Add to 90/180/360 degrees

The angle at the circumference added to the angle at the centre is equal to:

90^o if the angle is acute.

180^o if the angle at the centre is obtuse.

360^o if the angle at the centre is a reflex angle.

  • Halving and doubling

By remembering the theorem incorrectly, the student will double the angle at the centre, or half the angle at the circumference.

Top tip: look at the angles. The angle at the centre is always larger than the angle at the circumference (this isn’t so obvious when the angle at the circumference is in the opposite segment).

Angle at the centre is twice the angle at the circumference common misconception 1 1

  • Angles are the same

The angle at the circumference and the centre is incorrectly assumed to be equal.

Practice angle at the centre questions

1. ABCD is an arrowhead inscribed in a circle. The point B is the centre of the circle. Calculate the size of angle ABC , labelled \theta .

 

Practice Angle at the Centre question 1 1

36^o
GCSE Quiz False

18^o
GCSE Quiz False

144^o
GCSE Quiz True

108^o
GCSE Quiz False

72 \times 2=144^o (angle at the centre is twice the angle at the circumference)

2. Below is a circle with centre D. A, B, and C lie on the circumference of the circle where AB is the diameter of the circle. Calculate the size of angle ABC .

 

Practice Angle at the Centre question 2 1

236^o
GCSE Quiz False

59^o
GCSE Quiz True

62^o
GCSE Quiz False

160^o
GCSE Quiz False

118 \div 2=59^o (angle at the centre is twice the angle at the circumference)

3.  ABCD  is a cyclic quadrilateral. E is the centre of the circle. Calculate the size of the reflex angle at E .

Practice Angle at the Centre question 3 1

236^o
GCSE Quiz True

31^o
GCSE Quiz False

124^o
GCSE Quiz False

298^o
GCSE Quiz False

62\times2=124^o (angle at the centre is twice the angle at the circumference)

 

360 – 124 = 236^o (angles at a point)

4. A, B, and C are points on the circle with radius CD . Calculate the size of angle BAC .

 

Practice Angle at the Centre question 4 1

47^o
GCSE Quiz False

94^o
GCSE Quiz False

43^o
GCSE Quiz False

23.5^o
GCSE Quiz True

180-(74+59) = 47^o

 

47 \div 2=23.5^o (angle at the centre is twice the angle at the circumference)

5. A, B and C are points on the circle with centre D . EF and GH are tangents that meet outside the circle at the point P . Calculate the size of the angle ACD .

 

Practice Angle at the Centre question 5 1

96^o
GCSE Quiz False

114^o
GCSE Quiz True

132^o
GCSE Quiz False

264^o
GCSE Quiz False

APB = 48^o (vertically opposite)

 

ADB = 180 – 48 = 132^o

 

Reflex at ADB = 360 – 132 = 228^o (angles at a point)

 

228 \div 2=114^o (angle at the centre is twice the angle at the circumference)

6. The points A, B, and C lie on the circumference of a circle with centre O . The line DE is a tangent to the circle at point B . Calculate the size of angle AOB .

 

Practice Angle at the Centre question 6 1

36^o
GCSE Quiz False

72^o
GCSE Quiz False

108^o
GCSE Quiz False

144^o
GCSE Quiz True

ACB = ABE = 72^o (alternate segment theorem)

 

72 \times 2=144^o (angle at the centre is twice the angle at the circumference)

Angle at the centre GCSE questions

1. Calculate the size of angle BCD .

 

Angle at the centre GCSE question 1 1

 

(4 marks)

Show answer

Reflex angle at BCD = 123 \times 2=246^o

(1)

 

Angle at the centre is twice the angle at the circumference

(1)

 

Angle BCD = 360-246 = 114^o

(1)

 

Angles at a point total 360^o

(1)

2.

(a) A, B, C, D are points on the circumference of a circle with centre O . BD is the diameter of the circle. FG is a tangent at the point B and is parallel to the line AC . Angle ABF = 74^o . Calculate the size of angle BOC .

 

Angle at the centre GCSE question 2a 1

 

(b) Hence or otherwise, calculate the size of angle y.

 

Angle at the centre GCSE question 2b 1

 

(8 marks)

Show answer

(a)

 

BAC = 74^o

(1)

 

Alternate angles in parallel lines are the same

(1)

 

BOC = x= 74\times2 = 148^o

(1)

 

Angle at the centre is twice the angle at the circumference

(1)

 

(b)

 

COD = 180-148 = 32^o

(1)

 

Angles on a straight line total 180^o

(1)

 

ODC = COD so y=32^o

(1)

 

ODC is an isosceles triangle

(1)

3. The circle below has centre C. Points A, B, and D lie on the circumference. Calculate the size of angle BCD , marked \theta .

 

Angle at the centre GCSE question 3 1

 

(2 marks)

Show answer

BCD  = 66 \times 2=132^o

(1)

 

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

(1)

Learning checklist

You have now learned how to:

  • Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results

The next lessons are

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