Circle Theorems Explained: A Complete GCSE Guide

Circle theorems can feel like a long list of rules to memorise, but most questions follow the same pattern: identify the theorem, write down the fact, then use it to find a missing angle. This guide explains each theorem in plain language and shows you how examiners usually hide them in a diagram.

1. The angle at the centre is twice the angle at the circumference

If two points on the circumference are joined to the centre and to another point on the circumference, the angle at the centre is double the angle at the edge.

• Both angles must be drawn from the same arc
• If the angle at the circumference is 35°, the angle at the centre is 70°
• This is one of the most frequently tested theorems

Watch out for reflex angles. If the question asks for the larger angle at the centre, you may need to subtract your answer from 360°.

2. The angle in a semicircle is 90°

A triangle drawn inside a circle with one side as the diameter always has a right angle opposite that diameter.

• If you see a diameter, look for a right angle
• If you see a right angle on the circumference, the hypotenuse is probably the diameter
• This theorem is often combined with Pythagoras or trigonometry

3. Angles in the same segment are equal

Any two angles drawn from the same chord to the same side of the circumference are equal.

• Look for a shared base chord with two triangles sitting on the same arc
• The angles opposite that shared chord will be equal
• Exam questions often draw one obvious angle and one hidden angle on the same arc

4. Opposite angles of a cyclic quadrilateral add to 180°

A cyclic quadrilateral has all four vertices on the circumference. Its opposite angles are supplementary.

• Angle A + Angle C = 180°
• Angle B + Angle D = 180°
• If one angle is 110°, the opposite angle is 70°

Be careful: the quadrilateral must be cyclic. If one vertex is inside or outside the circle, the theorem does not apply.

5. The tangent is perpendicular to the radius

A tangent to a circle touches the circle at exactly one point. At that point, the radius and tangent meet at 90°.

• Draw the radius to the point of contact if it is not shown
• This creates a right-angled triangle you can solve
• Often used with Pythagoras to find lengths

6. Two tangents from a point are equal

If two tangents are drawn to a circle from the same external point, the lengths from that point to each point of contact are equal.

• This creates a pair of congruent right-angled triangles
• Useful for finding missing lengths or proving symmetry
• Remember the line from the external point to the centre bisects the angle between the tangents

7. The alternate segment theorem

The angle between a tangent and a chord is equal to the angle in the alternate segment.

• Find the angle between the tangent and any chord drawn from the point of contact
• The angle in the opposite segment will be equal
• This is the theorem students forget most often, so practise recognising the tangent-chord pair

8. The perpendicular from the centre bisects a chord

A line drawn from the centre of a circle perpendicular to a chord cuts the chord exactly in half.

• This creates two identical right-angled triangles
• Combine with Pythagoras to find the distance from the centre to the chord
• The reverse is also true: a line from the centre that bisects a chord is perpendicular to it

How to approach circle theorem questions

Most exam questions test more than one theorem at once. A reliable method is:

• Mark every angle and length you already know
• Look for radii, diameters, and tangents first — these give right angles
• Trace the arc that links the angle you want to a known angle
• Write a short reason next to each step, such as "angle in a semicircle" or "angles in same segment"

Examiners award marks for correct reasons, so even if you are unsure of the final number, state the theorem you are using.

Common mistakes to avoid

• Confusing the angle at the centre with the angle at the circumference
• Forgetting that a diameter creates a right angle
• Assuming a quadrilateral is cyclic without evidence
• Missing a tangent because it looks like a straight line continuing outside the circle
• Not giving a written reason when the question asks you to "give reasons"

When to get a tutor

If circle theorem questions always seem to need a diagram you cannot see, a tutor can teach you the visual patterns that reveal which theorem to use. A good GCSE maths tutor will show you how to annotate diagrams quickly and pick up method marks even when the final answer is tricky.

Browse maths tutors on TutorSite and filter by GCSE experience.

Final thought: circle theorems are about pattern recognition

You do not need to memorise every diagram. You need to recognise the triggers: a diameter, a tangent, a shared arc, or four points on a circle. Once you can spot the trigger, the theorem follows. Practise naming the theorem before you calculate the angle, and your confidence will grow quickly.