Playbook
Circles
69 q · 26% HARD · about 3.3 a paper. Tangents from an external point have never set a HARD question; the HARD is spread over the chord, circumcircle and touching-circle pages.
- Questions in the bank
- 69
- q/paper (21 papers)
- 3.29
- Tagged HARD
- 26%
- Subtopics
- 7
Strand: Selective
When you’ll see it
Chords, tangents, angles in a circle, or two circles touching.
How this chapter is tested
69 questions, about 3.3 a paper, at 26% HARD. Tangents from an external point have never set a HARD question; the HARD is spread over the chord, circumcircle and touching-circle pages.
Almost every question rests on two right triangles: radius, half-chord and distance from the centre; and radius, tangent and the line to the centre.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Chords
The perpendicular from the centre bisects a chord: r² = d² + (c/2)².
Angles in a circle
The angle at the centre is twice the angle at the circumference; opposite angles of a cyclic quadrilateral add to 180°.
Tangents
A tangent is perpendicular to the radius; the two tangents from a point are equal.
Power of a point
Intersecting chords: AP × PB = CP × PD; tangent² = product of the secant parts.
Two circles
Direct common tangent √(d² − (r₁ − r₂)²); touching externally d = r₁ + r₂.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.
Parallel chords
Same side of the centre: subtract the distances; opposite sides: add them. The question may not say which.
The reflex angle
An inscribed angle over 90° stands on the major arc; the central angle is 360° minus twice it.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each subtopic. Read the notes once, then drill subtopic by subtopic below.
Circles notesDrill every Circles question
69 questions from the bank, across 7 subtopics.
Drill one subtopic at a time
The 7 subtopics, in teaching order.
- Chords and PerpendicularsDrill Chords and Perpendiculars
- Angles in a CircleDrill Angles in a Circle
- Circumcircle and LocusDrill Circumcircle and Locus
- Tangents from an External PointDrill Tangents from an External Point
- Intersecting Chords and Power of a PointDrill Intersecting Chords and Power of a Point
- Common Tangents and Common ChordsDrill Common Tangents and Common Chords
- Touching CirclesDrill Touching Circles
Related playbooks
Often paired with this one — the technique or the trap overlaps. Drill these next.