Circle Theorems Higher
Draw any triangle with its corners on a circle and something remarkable happens: angles that look unrelated are locked together. Drag the points on each circle below and watch the measured angles change, while the relationship between them never breaks. Then learn to give the reasons in the exact words examiners want, and prove why the theorems are true.
GCSE subject content G9: identify and apply circle definitions and properties, including centre, radius, chord, diameter, circumference, tangent, arc, sector and segment. G10 (Higher): apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results.1 · The words you need
This builds on the KS3 page Circles, which names the parts of a circle. Circle theorems use a few extra words. Press a button to highlight each one, and drag the points A, B, P and Q round the circle.
The most important new idea is subtended. Take the chord or arc AB. Join its ends to another point: the angle you make "stands on" AB, or is subtended by AB. Joining A and B to the centre O gives the angle subtended at the centre; joining them to a point P on the circle gives the angle subtended at the circumference.
2 · Angles at the centre and in a semicircle Higher
Theorem 1. Drag A, B and P. The angle at the centre is always exactly twice the angle at the circumference, as long as both stand on the same arc.
Reason: the angle at the centre is twice the angle at the circumference.
Spot it in disguise. The shape can look like an arrowhead (O inside triangle APB), a "bird" or "bow tie" (O outside), or have P on the minor arc so that the angle at the centre is reflex (more than 180°). It still works: just make sure both angles stand on the same arc.
Theorem 2 is a special case. If AB is a diameter, the angle at the centre is a straight line, 180°, so the angle at the circumference is half of that:
Drag P, or drag A to turn the diameter.
Reason: the angle in a semicircle is 90°.
3 · Same segment and cyclic quadrilaterals Higher
Theorem 3. Angles at the circumference standing on the same arc are equal. Here P and Q both look at the chord AB from the same side (they're in the same segment). This shape is often called a "bow tie".
Reason: angles in the same segment are equal.
Theorem 4. A cyclic quadrilateral has all four corners on a circle. Its opposite angles always add up to 180°. (That's why, above, when P and Q were in different segments their angles added to 180°: APBQ was a cyclic quadrilateral.)
Reason: opposite angles in a cyclic quadrilateral add up to 180°.
Trap: the quadrilateral must have all four vertices on the circle. If one corner is the centre O, it is not cyclic and the theorem doesn't apply.
4 · Tangents Higher
A tangent touches the circle at exactly one point. Theorem 5: a tangent is perpendicular (at 90°) to the radius at the point where it touches. Tilt the line away from 90° and it cuts the circle a second time, so it is no longer a tangent.
Reason: a tangent is perpendicular to the radius at the point of contact.
Theorem 6: two tangents drawn from the same outside point are equal in length. Drag P. The shape OAPB is a kite: two right angles, two equal radii, two equal tangents, and OP is a line of symmetry.
Radius 5 cm. Drag P anywhere outside the circle.
Reason: tangents from an external point are equal in length.
5 · Chords and the centre Higher
Theorem 7: the perpendicular from the centre to a chord bisects the chord (cuts it into two equal halves). It works the other way too: the line from the centre to the midpoint of a chord meets it at 90°. Combine it with Pythagoras to find lengths.
Reason: the perpendicular from the centre to a chord bisects the chord.
6 · The alternate segment theorem Higher
Theorem 8 is the hardest to spot. A tangent touches the circle at A, and a chord AB goes from A across the circle. The angle between the tangent and the chord equals the angle that the chord subtends in the alternate (other) segment, at any point C on the far side of the chord.
Drag A, B or C. Look for a tangent and a chord meeting at the same point: then look in the other segment for the matching angle.
Reason: alternate segment theorem (the angle between a tangent and a chord is equal to the angle in the alternate segment).
7 · Giving reasons Higher
In exams, "give reasons for your answer" means every step needs a reason, written as a recognised fact. "Because it's a circle" or "it's 90" earns nothing; the full statement does. Typical wordings:
| Theorem | Exam wording |
|---|---|
| Centre | The angle at the centre is twice the angle at the circumference. |
| Semicircle | The angle in a semicircle is 90°. |
| Same segment | Angles in the same segment are equal. |
| Cyclic quadrilateral | Opposite angles in a cyclic quadrilateral add up to 180°. |
| Tangent and radius | A tangent is perpendicular to the radius at the point of contact. |
| Two tangents | Tangents from an external point are equal in length. |
| Chord | The perpendicular from the centre to a chord bisects the chord. |
| Alternate segment | The angle between a tangent and a chord equals the angle in the alternate segment. |
You'll also need the basic facts: angles in a triangle add up to 180°; angles on a straight line add up to 180°; base angles of an isosceles triangle are equal (and two radii make a triangle isosceles!). Find x, then choose the circle theorem you used.
Which circle theorem did you use?
8 · Proving the theorems Higher
Higher papers can ask you to prove a theorem. Every proof uses only facts you already know: radii are equal (so you get isosceles triangles), angles in a triangle add to 180°, and the exterior angle of a triangle equals the sum of the two opposite interior angles. Choose a theorem and step through its proof.
Worked examples
O is the centre. Angle APB = 38° (P on the major arc). Find angle AOB, giving a reason. Higher
AB is a diameter and P is on the circle. Angle PAB = 27°. Find angle PBA. Higher
ABCD is a cyclic quadrilateral with angle A = 104° and angle B = 71°. Find angles C and D. Higher
PA and PB are tangents from P to a circle with centre O. Angle APB = 48°. Find angle AOB and angle OAB. Higher
A chord of length 16 cm is 6 cm from the centre of a circle. Find the radius. Higher
TA is a tangent at A. Angle TAB = 65° and angle ABC = 58°, where C is on the major arc. Find angle ACB and angle BAC. Higher
Multi-step: O is the centre, A, B, C on the circle, angle OAB = 22°. Find angle ACB (C on the major arc). Higher
Quick check
What you've learned
- An angle is subtended by an arc or chord when its arms go to the arc's ends. A cyclic quadrilateral has all four vertices on the circle.
- 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 add up to 180°.
- A tangent is perpendicular to the radius at the point of contact; tangents from an external point are equal in length; the perpendicular from the centre to a chord bisects the chord.
- Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
- Give a reason for every step using the full wording. Look for radii: two radii make an isosceles triangle.
- The proofs use isosceles triangles, the exterior angle of a triangle, angle sums and congruent triangles (RHS).