Mathematics 0580 · Syllabus tracker
Circles
Circle vocabulary and the circle theorems: semicircles, tangents, angles at the centre, same segment, cyclic quadrilaterals, alternate segment and chords.
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Quick revision
Circles on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Circle vocabulary and every circle theorem, with the reasons examiners want.
1Circle words
- Radius: centre to edge. Diameter: across through the centre (2 × radius).
- Chord: a line joining two points on the circle. Tangent: a line touching the circle at one point.
- Arc: part of the circumference. Sector: a 'pizza slice'. Segment: area between a chord and the arc.
Worked example
What is the line from the centre to the point where a tangent touches?
- It is a radius.
A radius (it meets the tangent at 90°)
2Right angles in circles
- The angle in a semicircle is 90° (angle at the circumference standing on a diameter).
- A tangent meets the radius at 90°.
- Tangents from an external point are equal in length (look for an isosceles triangle or a kite).
- The perpendicular from the centre to a chord bisects the chord.
Worked example
AB is a diameter and C is on the circle. Angle CAB = 35°. Find angle CBA.
- Angle ACB = 90° (angle in a semicircle).
- 180 − 90 − 35.
55°
3Angles at the centre and circumference
- The angle at the centre is twice the angle at the circumference standing on the same arc.
- Angles in the same segment (standing on the same chord, same side) are equal.
- Opposite angles of a cyclic quadrilateral add to 180°.
Worked example
O is the centre. Angle AOC = 140°. B is on the major arc and D on the minor arc. Find angle ADC.
- Angle ABC = 70° (angle at the centre is twice the angle at the circumference).
- ABCD is cyclic, so angle ADC = 180 − 70.
110°
4Alternate segment theorem
- The angle between a tangent and a chord equals the angle in the alternate segment (on the other side of the chord).
- Spot it: a tangent, a chord from the point of contact, and a triangle drawn in the circle.
Worked example
A tangent at T makes 58° with chord TA. B is on the circle in the alternate segment. Find angle TBA.
- Alternate segment theorem.
58°
5Writing reasons
- Name the theorem every time: 'angle in a semicircle', 'angle at the centre is twice the angle at the circumference', 'opposite angles of a cyclic quadrilateral add to 180°', 'tangent perpendicular to radius', 'alternate segment theorem'.
- Radii are equal → isosceles triangles: use them to find base angles.
Where to start
- Mark radii and tangents on the diagram.
- Look for right angles first.
- Look for isosceles triangles.
- Then centre/circumference, same segment, cyclic quadrilateral, alternate segment.
Worked example
Tangents from P touch the circle at A and B. Angle APB = 50°. Find angle PAB.
- PA = PB (tangents from a point).
- Triangle PAB is isosceles: (180 − 50) ÷ 2.
65°
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Angle at circumference = 2 × angle at centre | It's the other way: centre = 2 × circumference |
| Cyclic quadrilateral rules with the centre as a corner | All four corners must be on the circle |
| Giving the angle without a reason | Each step needs the theorem named |
| Assuming a line is a diameter | Only if it goes through the centre |
★In the exam
- Diagrams are not to scale — use theorems, never measure.
- Find the easy angles first (right angles, isosceles), then the theorems.
- Learn the exact wording of each theorem for the 'give a reason' marks.