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.

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.

### What is the angle at the centre? ### Key parts of a circle needed for this theorem

• 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            Description

A, B, and D are points on the
circumference and C is the centre
of the circle.

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 ## 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 .

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

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 :

### 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.

Here we have:

• AC = BC = CD = Radii
• AB is a chord
• 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 .

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°

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

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:

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.

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)

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 .

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°

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

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:

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:

### 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 .

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 .

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

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).

• 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 . 36^o 18^o 144^o 108^o 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 . 236^o 59^o 62^o 160^o 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 . 236^o 31^o 124^o 298^o 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 . 47^o 94^o 43^o 23.5^o 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 . 96^o 114^o 132^o 264^o 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 . 36^o 72^o 108^o 144^o 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 . (4 marks)

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 . (b) Hence or otherwise, calculate the size of angle y. (8 marks)

(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 . (2 marks)

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

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