A complete guide to the circle theorems every learner needs — angle in a semicircle, the centre-circumference relationship, cyclic quadrilaterals, tangent properties and the alternate segment theorem.
A circle looks simple, but it hides a set of powerful angle relationships that let you find unknown angles without measuring anything. These circle theorems are some of the most tested ideas in geometry: once you recognise the shape a diagram is hinting at — a semicircle, a cyclic quadrilateral, a tangent — the correct theorem usually falls straight out. This guide works through every major circle theorem, from the basic vocabulary of a circle to the alternate segment theorem, with a diagram and worked example for each.
Before using any theorem, the vocabulary must be secure. The centre is the fixed middle point. A radius joins the centre to any point on the circle. A diameter is a chord through the centre, equal to two radii. A chord is any straight line joining two points on the circle. An arc is a part of the circumference; a sector is the pie-slice region between two radii and an arc; a segment is the region between a chord and an arc. A tangent is a straight line that touches the circle at exactly one point.
| Term | Definition |
|---|---|
| Radius | Centre to circumference |
| Diameter | Chord through the centre (2 × radius) |
| Chord | Line joining two points on the circle |
| Tangent | Line touching the circle at exactly one point |
| Arc | Part of the circumference |
| Sector | Region between two radii and an arc |
| Segment | Region between a chord and an arc |
If a triangle is drawn inside a circle so that one side is the diameter, the angle at the point touching the circumference is always exactly 90°. This is sometimes called Thales' theorem.
AB is a diameter of a circle, and C is a point on the circumference. If angle BAC = 35°, find angle ACB and angle ABC.
Angle ACB = 90° (angle in a semicircle). So angle ABC = 180 − 90 − 35 = 55°.
When an arc is viewed from the centre and from a point on the circumference (both on the same side of the arc), the angle at the centre is always twice the angle at the circumference.
An arc subtends an angle of 76° at the circumference. The angle subtended at the centre by the same arc is 2 × 76 = 152°.
Angles subtended by the same arc, from different points in the same segment of the circle, are always equal. This follows directly from the centre/circumference theorem, since every such angle equals half the same central angle.
Points C and D lie in the same segment of a circle, both viewing chord AB. If angle ACB = 48°, then angle ADB = 48° too.
A cyclic quadrilateral has all four vertices on the circumference of a circle. Its opposite angles always sum to 180°.
In cyclic quadrilateral ABCD, angle A = 105° and angle B = 80°. Find angles C and D.
Angle C = 180 − 105 = 75° (opposite angle A). Angle D = 180 − 80 = 100° (opposite angle B).
A tangent touches a circle at exactly one point, called the point of tangency. Two properties are essential: the tangent is always perpendicular (90°) to the radius drawn to the point of tangency, and two tangents drawn from the same external point are equal in length.
A tangent touches a circle at point T, and O is the centre. If OT = 5 cm and the tangent length from an external point P to T is 12 cm, find OP.
Since angle OTP = 90°, triangle OTP is right-angled. By Pythagoras: OP² = 5² + 12² = 25 + 144 = 169, so OP = 13 cm.
Where a tangent meets a chord at the point of tangency, the angle between the tangent and the chord equals the angle in the alternate segment — the angle subtended by that same chord on the far side of the circle.
A tangent meets a chord at a point on a circle, making an angle of 58° with the chord. The angle in the alternate segment is also 58°.
Exam questions typically combine two or three theorems in a single diagram. The reliable strategy is the same as with any angle-chasing problem: identify which theorem applies to each labelled feature (a diameter suggests the semicircle theorem; four points on a circle suggest a cyclic quadrilateral; a tangent suggests the alternate segment theorem or the perpendicular-radius property), then work through step by step.
AB is a diameter of a circle. C is a point on the circumference, and D is a point on the circumference such that ABCD is a cyclic quadrilateral with angle ADC = 115°. Find angle ACB and angle ABC if angle BAC = 25°.
Angle ACB = 90° (angle in a semicircle, since AB is a diameter). So angle ABC = 180 − 90 − 25 = 65°.
As a further check, angle ABC and angle ADC are opposite angles in the cyclic quadrilateral, so they should sum to 180°: 65 + 115 = 180 ✓.
Common errors include: applying the semicircle theorem when the given chord is not actually a diameter; confusing the centre-circumference relationship (doubling instead of halving, or vice versa); assuming opposite angles are equal rather than supplementary in a cyclic quadrilateral; and forgetting that the alternate segment theorem only applies at the exact point where the tangent touches the circle.
| Theorem | Statement |
|---|---|
| Angle in a semicircle | Always 90° |
| Angle at the centre | Twice the angle at the circumference (same arc) |
| Angles in the same segment | Equal |
| Cyclic quadrilateral | Opposite angles sum to 180° |
| Tangent and radius | Perpendicular (90°) at the point of tangency |
| Two tangents from a point | Equal in length |
| Alternate segment theorem | Tangent-chord angle equals the angle in the alternate segment |
Every circle theorem question is a search for which of these seven facts the diagram is hinting at. With practice, recognising the shape — a diameter, a cyclic quadrilateral, a tangent meeting a chord — becomes automatic.
This article is original EDUSAMBAM educational writing. It is designed as a broad circle theorems resource covering the core angle relationships needed for confident geometric reasoning with circles. Exact examination requirements can vary between examination boards and syllabuses, so students should also compare their work with the specification and past-paper requirements of their own board.
Recommended study approach: learn the seven theorems in the summary table with their diagrams, practise identifying which theorem a diagram is hinting at, then attempt multi-step problems that combine several theorems at once.
20 questions covering circle vocabulary, the semicircle and centre-circumference theorems, cyclic quadrilaterals, tangent properties and the alternate segment theorem. Answer every question, then submit to see your score instantly.