Teach Me Maths

Circle Theorems

Use circle theorems to find angles and lengths, combine them with algebra, and give precise reasons for each step.

Start with the circle

A chord joins two points on a circle. A diameter is a chord through the centre. A tangent touches the circle at one point. An arc is part of the circumference; a chord splits the circle into two segments.

For AQA Level 2 Further Maths, learn the eight theorems below. In an angle calculation or proof, name the theorem you use. Radii of the same circle are equal, so two radii also give an isosceles triangle.

Angle at the centre

The angle at the centre is twice the angle at the circumference standing on the same arc. Use the arc that does not contain the point on the circumference.

A reflex angle is greater than 180^\circ and less than 360^\circ. If C is on the minor arc AB, its angle stands on the major arc, so use the reflex angle at O. If only the minor central angle is given, subtract it from 360^\circ first.

Worked example 1

O is the centre. The minor angle AOB is 104^\circ, and C lies on the major arc AB. Find \angle ACB.

  2. \angle AOB=2\angle ACB because the angle at the centre is twice the angle at the circumference on the same arc.

  3. Your turn. Calculate 104\div2

    \angle ACB=52^\circ

  4. Answer: 52^\circ. State the theorem as well as the calculation.

Worked example 2

O is the centre. The minor angle AOB is 124^\circ, and C lies on the minor arc AB. Find \angle ACB.

  2. The reflex angle AOB=360-124=236^\circ

  3. Your turn. Calculate 236\div2

    \angle ACB=118^\circ

  4. Answer: 118^\circ. Halving 124^\circ would use the wrong arc.

Angles in the same segment

Angles at the circumference standing on the same chord, with their vertices in the same segment, are equal. Check that the vertices lie on the same side of the chord.

Worked example 3

C and D lie on the same side of chord AB. Given \angle ADB=38^\circ and \angle ACB=(3x+2)^\circ, find x. All four points lie on the circle.

  2. Angles in the same segment are equal, so 3x+2=38

  3. Your turn. Find x

    3x=36, so x=12

  4. Answer: x=12. The angle itself is 38^\circ, not 12^\circ

Angle in a semicircle

An angle at the circumference standing on a diameter is 90^\circ. The right angle is opposite the diameter. This is the centre-angle theorem with a central angle of 180^\circ.

Worked example 4

AB is a diameter and C lies on the circle. Given \angle BAC=31^\circ, find \angle ABC.

  2. Your turn. What is \angle ACB in degrees?

    The angle in a semicircle is 90^\circ

  3. \angle ABC=180-90-31=59^\circ, using the angle sum of a triangle.
  4. Answer: 59^\circ

Opposite angles in a cyclic quadrilateral

A cyclic quadrilateral has all four vertices on the circumference of one circle. Its opposite angles add to 180^\circ. This rule concerns opposite angles, not neighbouring ones.

Worked example 5

ABCD is cyclic, with opposite angles \angle DAB=(2x+15)^\circ and \angle BCD=(3x-10)^\circ. Find x and both angles.

  2. (2x+15)+(3x-10)=180, since opposite angles of a cyclic quadrilateral sum to 180^\circ

  3. Your turn. Solve 5x+5=180

    x=35

  4. Answer: x=35, \angle DAB=85^\circ and \angle BCD=95^\circ. Check: 85+95=180

Radius perpendicular to a tangent

The radius to the point of contact is perpendicular to the tangent. Mark the 90^\circ angle where the radius meets the tangent; joining the centre to the contact point often reveals a right-angled triangle.

Worked example 6

PA is a tangent to a circle with centre O, touching at A. Given \angle AOP=(2x+8)^\circ and \angle OPA=(x+13)^\circ, find x and both angles.

  2. \angle OAP=90^\circ because a radius is perpendicular to the tangent at the point of contact. The other two angles therefore add to 90^\circ

  3. Your turn. Solve (2x+8)+(x+13)=90

    3x+21=90, so 3x=69 and x=23

  4. Answer: x=23, \angle AOP=54^\circ and \angle OPA=36^\circ. Check: 54+36+90=180

Equal tangents from one external point

The two tangents from the same external point to a circle are equal in length. Each radius to a contact point is perpendicular to its tangent, giving two right-angled triangles.

Worked example 7

Tangents PA and PB touch a circle with centre O. Given \angle APB=64^\circ, PA=x+3 and PB=2x-4, find the minor angle AOB and the tangent lengths.

  2. \angle OAP=\angle OBP=90^\circ. The angles of quadrilateral OAPB sum to 360^\circ, so \angle AOB=360-90-90-64=116^\circ

  3. Your turn. Equal tangents give x+3=2x-4. Find x

    x=7

  4. Answer: the minor angle AOB is 116^\circ and PA=PB=10 units.

The alternate segment theorem

The angle between a tangent and a chord equals the angle at the circumference in the alternate segment. Use the same chord in both angles, and choose the circumference vertex on the opposite side of the chord from the tangent–chord angle.

Worked example 8

The tangent at A makes an angle of 62^\circ with chord AB, on the side of AB opposite C. Points A,B,C lie on the circle and \angle BAC=47^\circ. Find \angle ABC.

  2. Your turn. What is \angle ACB in degrees?

    \angle ACB=62^\circ by the alternate segment theorem.

  3. \angle ABC=180-47-62=71^\circ
  4. Answer: 71^\circ

Perpendicular from the centre to a chord

The perpendicular from the centre of a circle to a chord bisects the chord: it cuts it into two equal lengths. The converse also holds: a line from the centre to the midpoint of a chord which is not a diameter is perpendicular to the chord. A radius and half the chord form two sides of a right-angled triangle.

Worked example 9

A circle has radius 13 cm and chord AB of length 24 cm. The perpendicular from its centre O meets AB at M. Find OM.

  2. AM=24\div2=12 cm. Triangle OMA is right-angled at M, with hypotenuse OA=13 cm.

  3. Your turn. Calculate OM^2=13^2-12^2

    OM^2=25

  4. Answer: OM=5 cm. Use the positive root because this is a length.

Common mistakes

  • Using angles on different arcs. Identify the same arc or chord before applying the theorem.
  • Halving the wrong central angle. Check whether the circumference angle uses the minor or reflex central angle.
  • Adding adjacent cyclic angles. The opposite angles of a cyclic quadrilateral add to 180^\circ.
  • Giving only “circle theorem” as a reason. Name the specific angle or length relationship used.
  • Assuming a chord is a diameter. Check that it is given or proved to pass through the centre.

Now try it: Circle Theorems practice questions

More on this topic: Circle Theorems worksheet with full solutions