Angles in a circle
Circle theorems relate angles formed by chords, radii and tangents. They let you find unknown angles without measuring. Learn each rule and the diagram that goes with it.
Key idea
The angle at the centre is twice the angle at the circumference standing on the same arc.
The angle in a semicircle is 90°.
Angles in the same segment are equal.
Cyclic quadrilaterals and tangents
In a cyclic quadrilateral (all four vertices on the circle), opposite angles add to 180°. A tangent meets a radius at 90°, and two tangents drawn from the same external point are equal in length.
Worked example
An angle at the circumference standing on an arc is 40°. Find the angle at the centre on the same arc.
Angle at centre = 2 × angle at circumference = 2 × 40° = 80°.
If instead the opposite angle of a cyclic quadrilateral were 70°, its partner would be 180° - 70° = 110°.
Alternate segment theorem
The angle between a tangent and a chord equals the angle in the alternate segment (the angle subtended by that chord on the far side of the circle).
Remember
- Centre angle = 2 × circumference angle on the same arc.
- Semicircle angle = 90°; cyclic quad opposite angles sum to 180°.
- Tangent is perpendicular to the radius at the point of contact.