Circle theorems
Eight circle theorems, one at a time. Drag the points round the circle and watch the angles change while the theorem keeps holding, then learn the reason you have to write in the exam.
Readouts
What's happening
Every circle theorem comes from one fact: the radii OA, OB and OC are all equal, so the triangles they make with chords are isosceles. Splitting the figure at the centre turns that into the big one: the angle at the centre is twice the angle at the circumference standing on the same arc. The rest follow. Two angles standing on the same arc both equal half the same centre angle, so they are equal. A diameter makes a centre angle of 180°, so the angle in a semicircle is 90°. Opposite angles of a cyclic quadrilateral stand on arcs that make up the whole circle, so they add to 180°. A tangent touches the circle at one point and is perpendicular to the radius there, which makes the two tangents from an outside point equal (congruent right-angled triangles) and gives the alternate segment theorem. The perpendicular from the centre to a chord splits an isosceles triangle in half, so it bisects the chord.
GCSE Maths (AQA, Edexcel, OCR) Higher: circle theorems, including the reasons required in proofs. Also IGCSE Maths and Scottish National 5.
Work through the numbers with Scientific Calculator.
Challenge
Predict first: on the Centre theorem, put A at 40°, B at 150° and C at 270°. What is angle x = ∠ACB? Work out ∠AOB first, type x in the box (the figure hides x while you type) and press Check.
∠AOB = 150° − 40° = 110°, and the angle at the circumference is half of it, so x = 55°. Now drag C onto the small arc between A and B: ∠AOB becomes the reflex angle 250° and x becomes 125°, still exactly half.
FAQ
- What are the circle theorems for GCSE?
- The angle at the centre is twice the angle at the circumference; angles in the same segment are equal; the angle in a semicircle is 90°; opposite angles in a cyclic quadrilateral add to 180°; a tangent is perpendicular to the radius; tangents from an external point are equal; the alternate segment theorem; and the perpendicular from the centre to a chord bisects the chord.
- How do I write the reason in an exam?
- Quote the theorem in words next to each step, for example 'angle at the centre is twice the angle at the circumference' or 'opposite angles in a cyclic quadrilateral add up to 180°'. Most mark schemes need the reason as well as the number.
- What is the alternate segment theorem?
- The angle between a tangent and a chord is equal to the angle in the alternate segment, that is, the angle the chord makes at any point on the circle on the other side of the chord.
- Does the centre theorem work for a reflex angle?
- Yes. If C is on the minor arc, the angle at the centre on the same arc is reflex, and it is still twice ∠ACB. That is why ∠ACB can be more than 90°.