GCSE Revision

Circle theorems

Angles at the centre and the circumference (Higher tier only)

Learn each theorem in this wording: it's the reason you give.

  1. Draw two lines from the ends of an arc to the centre, and two to a point on the circumference: the angle at the centre is double.
  2. The angle in a semicircle is 90°. A triangle drawn on a diameter, with its third corner on the circle, has a right angle at that corner.
  3. Angles drawn from the ends of a chord to points on the same side of the circle are equal.
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Example: the angle at the centre, standing on arc AB, is 130°. Find the angle at the circumference standing on the same arc.

  1. The angle at the centre is twice the angle at the circumference
  2. 130 ÷ 2 = 65°

Key terms

Cyclic quadrilaterals (Higher tier only)

A has all four corners on a circle.

  1. Opposite angles of a cyclic quadrilateral add up to 180°.

Example: ABCD is a cyclic quadrilateral. Angle ABC = 105° and angle BCD = 68°. Find angles ADC and DAB.

  1. Angle ADC is opposite angle ABC: 180 − 105 = 75°
  2. Angle DAB is opposite angle BCD: 180 − 68 = 112°

Check: 105 + 68 + 75 + 112 = 360, as for any quadrilateral.

Common mistake

It's opposite angles that add up to 180°, not angles next to each other.

Key terms

Tangents and chords (Higher tier only)

  1. At the point where a tangent touches the circle, the radius is perpendicular to it.
  2. Two tangents from the same point are equal in length. From a point outside the circle, the two tangents to the circle are the same length.
  3. A line from the centre that meets a chord at 90° cuts it in half.

Example: a tangent from point P touches a circle with centre O at A. OA = 5 cm and OP = 13 cm. Find PA.

  1. A tangent and a radius meet at 90°, so triangle OAP has a right angle at A
  2. Pythagoras: PA² = 13² − 5² = 169 − 25 = 144
  3. PA = 12 cm

Key terms

The alternate segment theorem (Higher tier only)

Picture a tangent touching the circle at A, and a chord AB. The angle between the tangent and AB equals any angle at the circumference standing on AB, at a point C on the other side of the chord.

Example: a tangent touches a circle at A. The angle between the tangent and chord AB is 70°. C is a point on the circle on the other side of AB. Find angle ACB.

  1. The angle between a tangent and a chord is equal to the angle in the alternate segment
  2. Angle ACB = 70°

Exam tip

Write the theorem's full wording as your reason. "Alternate segment" alone may not get the mark.

Using circle theorems (Higher tier only)

In longer problems, combine circle theorems with ordinary angle facts. Give a reason for every step.

Example: AB is a diameter of a circle with centre O. C is a point on the circle, and angle CAB = 32°. Find angle ABC.

  1. Angle ACB = 90°, because the angle in a semicircle is 90°
  2. Angle ABC = 180 − 90 − 32 = 58°, because angles in a triangle add up to 180°

Example: O is the centre of a circle, and A and B are points on it, with angle AOB = 100°. Find angle OAB.

  1. OA = OB, because they are both radii, so triangle OAB is isosceles
  2. Angle OAB = (180 − 100) ÷ 2 = 40°, because base angles of an isosceles triangle are equal

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