The radius joins the centre to a point on the circle. A diameter is a chord through the centre and has length 2r.
A chord is a segment joining two circle points. A tangent line meets the circle at exactly one point.
An arc is part of the circumference. A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc.
For radius r, circumference = 2πr and area = πr². With θ in degrees, arc length = (θ/360) × 2πr and sector area = (θ/360) × πr².
A 60° sector of radius 6 cm
Circle theorems (the key ones)
The angle at the centre is twice the angle at the circumference subtended by the same arc.
Angles in the same segment are equal.
The angle in a semicircle is 90°.
Opposite angles of a cyclic quadrilateral add up to 180°.
A tangent is perpendicular to the radius at the point of contact.
Tangents from an external point to a circle are equal in length.
The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
A line from the centre perpendicular to a chord bisects the chord.
Angle subtended by a diameter
Giving reasons
In angle problems write the angle and the theorem as a reason: "angle at centre is twice the angle at circumference".
Mark equal radii: triangles with two radii are isosceles.
The radius and tangent at contact
The radius to a tangent’s contact point is perpendicular to the tangent. In triangle OAT, O is the centre and A is the contact point, so angle OAT = 90°.
Tangent segments from the same external point to the two contact points have equal lengths.
A radius meets a tangent at 90°
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