Master all circle theorems for GCSE Maths with clear diagrams, explanations, and exam tips. Download as PDF for offline revision.
The angle at the centre is twice the angle at the circumference when subtended by the same arc.
When two points on a circle are connected to both the centre and a point on the circumference, the angle at the centre is always double the angle at the circumference.
Angles subtended by the same arc at the circumference are equal.
Any angles drawn from the same chord to points on the same side of the circumference will always be equal.
The angle in a semicircle is always 90° (a right angle).
When a triangle is drawn inside a semicircle with the diameter as its base, the angle at the circumference is always 90°. This is also known as Thales' theorem.
Opposite angles in a cyclic quadrilateral add up to 180°.
A cyclic quadrilateral has all four vertices on the circumference of a circle. The opposite angles always sum to 180°.
A tangent to a circle is perpendicular to the radius at the point of contact.
Where a tangent touches the circle, it meets the radius at exactly 90°. The radius is drawn from the centre to the point where the tangent touches.
Two tangents drawn from the same external point are equal in length.
If you draw two tangent lines from a point outside the circle, both tangent segments will have exactly the same length.
The angle between a tangent and a chord equals the angle in the alternate segment.
The angle formed between a tangent and a chord at the point of contact is equal to the inscribed angle subtended by the chord on the opposite side.
A perpendicular line from the centre of a circle to a chord bisects the chord.
If you draw a line from the centre perpendicular to a chord, it will cut the chord exactly in half. This works the other way too - a line from the centre to the midpoint of a chord is perpendicular to it.
The 8 circle theorems are: (1) Angle at the centre is twice the angle at the circumference, (2) Angles in the same segment are equal, (3) Angle in a semicircle is 90°, (4) Opposite angles in a cyclic quadrilateral sum to 180°, (5) Tangent meets radius at 90°, (6) Tangents from an external point are equal, (7) Alternate segment theorem, (8) Perpendicular from centre bisects chord.
The angle at the centre theorem and the angle in a semicircle are among the most frequently tested. The alternate segment theorem is often considered the trickiest and appears in higher-tier papers.
Look for key features: tangent lines (touching circle once), chords, diameters, angles at the centre vs circumference, or four points on the circle (cyclic quadrilateral). Drawing extra lines like radii often helps reveal which theorem applies.
A chord is a line segment with both endpoints on the circle's circumference, passing through the interior. A tangent is a line that touches the circle at exactly one point without crossing it.
This is a special case of the angle at centre theorem. The diameter creates an angle of 180° at the centre (a straight line), so the angle at the circumference is half of that: 180° ÷ 2 = 90°. This is also called Thales' theorem.
When a tangent and chord meet at a point on the circle, the angle between them equals the angle made by the chord in the 'other' (alternate) segment of the circle. Think of it as the angle 'reflecting' across the chord.
Both angles are subtended by the same arc. Using the angle at centre theorem, each angle at the circumference is half the angle at the centre for that arc. Since they're both half of the same central angle, they must be equal.
A cyclic quadrilateral has all four vertices (corners) lying on the circumference of a single circle. The key property is that opposite angles always add up to 180°.
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