Circle Theorems

The method, step by step

1. 1

   Angle at the centre is twice the angle at the circumference

   central angle 130° → circumference angle 65°

   Both angles must stand on the same arc. This is the theorem most others derive from, and the commonest error is applying it to two angles subtended by different arcs, which the diagram does not always make obvious.

2. 2

   The angle in a semicircle is 90°

   diameter as the base → the opposite angle is a right angle

   This is the previous theorem with a central angle of 180°, halved. Spotting a diameter in a diagram is therefore worth doing first, because it hands you a right angle for free and often unlocks Pythagoras or trigonometry.

3. 3

   Angles in the same segment are equal

   two angles on the same arc, same side → equal

   Any two angles at the circumference standing on the same arc are equal, because each is half the same central angle. Again the arc has to match — angles that look similar but stand on different arcs are not equal.

4. 4

   Opposite angles in a cyclic quadrilateral sum to 180°

   a + c = 180° and b + d = 180°

   A cyclic quadrilateral has all four vertices on the circle. Checking that every vertex genuinely touches the circumference matters, because the theorem fails for a quadrilateral that merely looks inscribed.

5. 5

   A tangent meets a radius at 90°

   tangent ⟂ radius at the point of contact

   This produces right-angled triangles wherever a tangent appears, which is usually the intended route into the question. The alternate segment theorem — the angle between tangent and chord equals the angle in the alternate segment — completes the set.

The theorems, stated as examiners expect

There are seven commonly examined circle theorems, and the wording matters as much as the geometry. Mark schemes accept the standard phrasings and often reject paraphrases, so learning the sentence is part of learning the theorem.

Written out: the angle at the centre is twice the angle at the circumference; the angle in a semicircle is 90°; angles in the same segment are equal; opposite angles in a cyclic quadrilateral sum to 180°; a tangent is perpendicular to the radius at the point of contact; tangents from an external point are equal in length; and the alternate segment theorem.

• Centre angle = 2 × circumference angle (same arc)
• Angle in a semicircle = 90°
• Angles in the same segment are equal
• Cyclic quadrilateral: opposite angles sum to 180°
• Tangent ⟂ radius at the point of contact
• Two tangents from a point are equal in length
• Alternate segment theorem

Nearly every error is the wrong arc

Most of these theorems carry a condition about which arc the angles stand on, and diagrams are drawn to make that condition easy to overlook. Two angles at the circumference are only equal if they subtend the same arc; a central and circumference angle only halve if they do too.

The fix is mechanical: before applying any theorem, trace the arc each angle stands on with a finger and confirm they match. It takes seconds and it eliminates the majority of wrong answers in the topic.

How to find the way in

Circle theorem questions are usually chains of two or three theorems, and the difficulty is finding the first one. Three things are worth looking for immediately: a diameter, which gives a right angle; a tangent, which gives another; and a quadrilateral with all four vertices on the circle.

Any of those three hands you an angle without further reasoning, and from there the remaining steps are normally straightforward. Students who scan for these features first solve these questions far more reliably than those who start from the angle they were asked for.

How we teach circle theorems

We require the exact wording in every written reason, because that is what the mark scheme accepts. Students who write 'because it's a semicircle' have the geometry right and will not always be given the reasoning mark.

We also teach the scan — diameter, tangent, cyclic quadrilateral — as an explicit first step. Circle theorem questions reward pattern recognition far more than they reward calculation, and giving students something specific to search for changes their success rate on them noticeably.