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In order to access this I need to be confident with:

Parts of a circle Angles in polygons Angles on a straight line Angles around a point Angles in parallel lines

Properties of triangles

Factorising

This topic is relevant for:

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?

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

      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

Locate the key parts of the circle for the theorem
Use other angle facts to determine the angle at the centre or the angle at the circumference
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 worksheet

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

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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.

DOWNLOAD FREE

Tangent of a circle is one of 7 circle theorems you will need to know. You may find it helpful to start with our main circle theorems page and then look in detail at the rest.

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

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 $BAD$ :

$BAD = 150 \div 2$

$BAD = 75°$

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

Locate the key parts of the circle for the theorem.

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 $BCD$ $= \theta$

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

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

Use 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 circumference $BAD$ , we need to multiply this number by $2$ to get the angle $BCD$ at the centre:

$BCD$ = $46$ × $2$

$BCD = 92°$ $\begin{matrix} \end{matrix}$

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

Locate the key parts of the circle for the theorem.

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 $AEC$ $= \theta$

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

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

Use 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 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

Locate the key parts of the circle for the theorem.

Here we have:

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

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

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 $= 180^o$ .

$BAD = 180 - (104 + 39)$

$BAD$ = $37°$

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

Use 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 circumference $BAC$ , we need to multiply this number by $2$ to get the angle $BCD$ at the centre:

$BCD = 37 \times 2$

$BCD = 74°$

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

Locate the key parts of the circle for the theorem.

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 $BAD$ $= \theta$

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

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

Use 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 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

Locate the key parts of the circle for the theorem.

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 $BOC$ $= \theta$

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

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

Use 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 circumference $BAC$ , we need to multiply this number by $2$ to get the angle $BOD$ at the centre:

$BOC$ = $82 \times 2$

$BOC$ = $164 °$

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

36^o

18^o

144^o

108^o

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

236^o

59^o

62^o

160^o

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

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)

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)

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)

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 $BCD$ .

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)

(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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