Circle Theorems

GCSE Maths · Geometry

Circle Theorems (Higher)

Circle theorems are rules about angles formed by chords, tangents, radii and arcs in a circle. You must know these for the Higher tier.

Key Terms

Chord: a straight line joining two points on the circumference
Tangent: a line that touches the circle at exactly one point
Arc: a section of the circumference
Sector: the "pizza slice" area between two radii and an arc
Segment: the region between a chord and an arc

The Theorems

1. Angle at the Centre

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

If the angle at the circumference is x, the angle at the centre is 2x.

2. Angle in a Semicircle

The angle in a semicircle is always 90°. (This is a special case of Theorem 1 — the angle at the centre is 180°, so the angle at the circumference is 90°.)

3. Angles in the Same Segment

Angles subtended by the same arc at the circumference are equal.

4. Cyclic Quadrilateral

A cyclic quadrilateral has all four vertices on the circumference. Opposite angles sum to 180°.

5. Tangent and Radius

A tangent to a circle is perpendicular to the radius at the point of contact (they meet at 90°).

6. Two Tangents from an External Point

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

7. Alternate Segment Theorem

The angle between a tangent and a chord equals the angle in the alternate segment.

8. Perpendicular from Centre to Chord

The perpendicular from the centre to a chord bisects the chord (cuts it in half).

Worked Examples

Worked Example 1: O is the centre. Angle AOB = 124°. Find angle ACB where C is on the major arc.

Angle ACB = 124° ÷ 2 = 62° (angle at centre = 2 × angle at circumference)

Worked Example 2: ABCD is a cyclic quadrilateral. Angle A = 73° and angle B = 105°. Find angles C and D.

Angle C = 180° − 73° = 107° (opposite angles in cyclic quadrilateral)
Angle D = 180° − 105° = 75°

Worked Example 3: A tangent meets the circle at P. The chord PQ makes an angle of 40° with the tangent. Find the angle PRQ where R is on the major arc.

Angle PRQ = 40° (alternate segment theorem)

Worked Example 4: A tangent from point T touches the circle at A. TA = 12 cm and the distance from T to the centre O is 13 cm. Find the radius.

Angle OAT = 90° (tangent perpendicular to radius)
By Pythagoras: OA² + 12² = 13²
OA² = 169 − 144 = 25
Radius = 5 cm

Proving Circle Theorems

Exam questions sometimes ask you to prove a theorem. You may use:

Isosceles triangles (two radii form equal sides)
Angle sum of a triangle (180°)
Previously proved theorems

Example proof structure (angle in a semicircle): Draw two radii to the circumference point. Each triangle formed is isosceles. Set up base angles, use the angle sum, and show the angle equals 90°.

Exam Tips

Always state the theorem name as your reason — "angle at centre is twice angle at circumference" earns marks
Look for the centre point O — if it is marked, Theorems 1, 5 and 8 are likely relevant
Look for tangent lines — if present, consider Theorems 5, 6 and 7
If four points lie on a circle, check for a cyclic quadrilateral
Draw extra lines (radii, chords) if they help you see the relationships