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Geometry

Circle theorems

All eight circle theorems on the GCSE specification, each with the rule as an examiner expects it stated and a worked example. Learn the reason as well as the result: in most circle theorem questions the reason is worth its own mark.

Foundation and HigherAQA · Edexcel · OCR · Eduqas

The eight theorems

  1. 1

    Angle at the centre

    The angle at the centre is twice the angle at the circumference when both are subtended by the same arc.

    ExampleIf the angle at the circumference is 35°, the angle at the centre is 70°.

  2. 2

    Angle in a semicircle

    The angle in a semicircle is always 90°. Any angle inscribed in a semicircle, with the diameter as its base, is a right angle.

    ExampleIf AB is a diameter and C sits on the circle, angle ACB = 90°.

  3. 3

    Angles in the same segment

    Angles subtended by the same chord in the same segment are equal.

    ExampleIf angles ADB and ACB both sit on chord AB in the same segment, then ADB = ACB.

  4. 4

    Cyclic quadrilateral

    Opposite angles in a cyclic quadrilateral add up to 180°, so they are supplementary.

    ExampleIf angle A = 110°, the opposite angle C = 70°.

  5. 5

    Tangent and radius

    A tangent to a circle is perpendicular to the radius at the point of contact.

    ExampleIf OA is a radius and TA is a tangent at A, then angle OAT = 90°.

  6. 6

    Two tangents from a point

    Two tangents drawn from the same external point to a circle are equal in length.

    ExampleIf PA and PB are tangents from point P, then PA = PB.

  7. 7

    Alternate segment theorem

    The angle between a tangent and a chord equals the inscribed angle on the opposite side of the chord.

    ExampleThe angle between the tangent and chord AB equals the angle in the alternate segment.

  8. 8

    Perpendicular from centre to chord

    The perpendicular from the centre of a circle to a chord bisects the chord.

    ExampleIf OM is perpendicular to chord AB, then AM = MB.

Where marks get lost

  • Name the theorem you used. Most circle theorem questions carry a mark for the reason, not just the number, so "angles in the same segment are equal" is worth writing out in full.
  • Look for the diameter first. If a triangle has the diameter as one side and its third point on the circle, you already have a 90° angle.
  • Mark equal angles and equal lengths on the diagram as you find them. Circle theorem questions usually chain, and the second step is much easier to see once the first is drawn on.
  • Check whether a quadrilateral is actually cyclic before using the 180° rule. All four vertices have to sit on the circle.

Questions students ask

How many circle theorems do I need for GCSE?

Eight, and they are all listed above. Higher tier papers can combine two or three in a single question, and the alternate segment theorem is Higher only.

Do I have to give a reason?

Yes. Circle theorem questions almost always award a mark for the reason. Writing "angle at the centre is twice the angle at the circumference" earns that mark even if you then make an arithmetic slip.

What is the alternate segment theorem in plain English?

Where a tangent touches a circle and a chord leaves from that same point, the angle squeezed between them equals the angle you would see standing in the segment on the other side of that chord.

Are circle theorems Foundation or Higher?

Most appear on both tiers, but the alternate segment theorem and the harder multi-step proofs are Higher only.

Reading it is not the same as being able to do it

Every rule above has practice questions behind it that regenerate with new numbers, marked the way the exam marks them.

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