IGCSE Maths · Geometry · Core & Extended

Circle Theorems: A Toolkit of Angle Shortcuts

Circle theorems are fixed relationships that always hold, no matter how the circle is drawn — once you recognise which theorem a diagram is showing, the angle just falls out. Core covers two theorems (semicircle, tangent-radius); every other theorem in this note is Extended-only, tagged throughout. This closes out the Geometry unit.

1. Angle in a Semicircle = 90°

If a triangle is drawn inside a circle so that one side is a diameter, the angle opposite that diameter is always exactly 90° — no matter where the third point sits on the circle.

Diagram and worked example A B C AB is a diameter of the circle. C is any other point on the circle. Find angle ACB.
Since AB is a diameter, angle ACB = 90° (angle in a semicircle).
Try it — Angle in a Semicircle
  1. AB is a diameter. C is a point on the circle. If angle BAC = 35°, find angle ABC (hint: use the triangle angle sum with the 90° you already know).
  2. Explain why the angle in a semicircle is always 90°, regardless of where the third point is placed on the circle.
Show answers
  1. 180 − 90 − 35 = 55°
  2. It’s a fixed circle theorem — however the point moves around the circle, the angle it makes with the two ends of the diameter always works out to exactly 90°.

2. Angle Between Tangent and Radius = 90°

A tangent touches a circle at exactly one point — and at that point, it is always perpendicular to the radius.

Diagram and worked example O T O is the centre, T is the point where the tangent touches the circle. Angle OTZ (between the radius OT and the tangent line) = 90°, always.
Try it — Tangent and Radius
  1. A radius meets a tangent at point T. What is the angle between them?
  2. A tangent and a chord meet at point T, and the angle between them (on one side) is 55°. Combined with the tangent-radius rule, find the angle between the radius and the chord.
Show answers
  1. 90°
  2. 90 − 55 = 35°

3. Angle at the Centre = Twice the Angle at the Circumference Extended only

For the same arc, the angle formed at the centre of the circle is always exactly double the angle formed at any point on the circumference (on the major arc).

Diagram and worked example O A B C 120° 60° O is the centre. Angle AOB = 120°. Find angle ACB.
Angle at the centre is double the angle at the circumference: angle ACB = 120 ÷ 2 = 60°
Try it — Centre and Circumference Extended
  1. Angle AOB (at the centre) = 150°. Find angle ACB (at the circumference).
  2. Angle ACB (at the circumference) = 42°. Find angle AOB (at the centre).
Show answers
  1. 150 ÷ 2 = 75°
  2. 42 × 2 = 84°

4. Angles in the Same Segment Are Equal Extended only

Any two angles subtended by the same chord, from points on the same arc, are equal to each other.

Diagram and worked example A B C D C and D are both on the same arc (the major arc, on the far side of chord AB). Angle ACB = 60°. Find angle ADB.
Both angles are in the same segment, subtended by the same chord AB: angle ADB = 60°
Try it — Same Segment Extended
  1. C and D are on the same arc, both looking at chord AB. Angle ADB = 47°. Find angle ACB.
  2. Why must angles in the same segment be equal? (Hint: connect this to the centre/circumference theorem.)
Show answers
  1. 47° (angles in the same segment are equal)
  2. Both angles are subtended by the same arc AB, so both equal half of the same angle at the centre — making them automatically equal to each other.

5. Cyclic Quadrilaterals: Opposite Angles Sum to 180° Extended only

A cyclic quadrilateral has all four vertices on the same circle. Its opposite angles are always supplementary (sum to 180°).

Diagram and worked example A B C D ABCD is a cyclic quadrilateral. Angle A = 85°. Find angle C.
Opposite angles sum to 180°: angle C = 180 − 85 = 95°
Try it — Cyclic Quadrilaterals Extended
  1. In a cyclic quadrilateral, angle B = 102°. Find angle D.
  2. A cyclic quadrilateral has angles A = 70°, B = 95°, C = x°, D = y°. Find x and y.
Show answers
  1. 180 − 102 = 78°
  2. x = 180−70 = 110° (opposite A). y = 180−95 = 85° (opposite B)

6. The Alternate Segment Theorem Extended only

The angle between a tangent and a chord drawn from the point of contact equals the angle in the alternate segment — the segment on the other side of the chord.

Diagram and worked example A B C 50° The tangent touches the circle at A. Chord AB is drawn, and the angle between the tangent and AB (marked, on the lower side) is 50°. C is a point in the alternate segment. Find angle ACB.
By the alternate segment theorem: angle ACB = 50°
Try it — Alternate Segment Theorem Extended
  1. A tangent meets a chord at 63°. Find the angle in the alternate segment.
  2. The angle in the alternate segment is 38°. Find the angle between the tangent and the chord.
Show answers
  1. 63° (the alternate segment theorem states they’re equal)
  2. 38°

7. Equal Chords Are Equidistant from the Centre Extended only

If two chords in a circle are the same length, they are always the same perpendicular distance from the centre.

Diagram and worked example O Chord AB and chord CD are the same length. The perpendicular distances from O to each chord are shown dashed — since AB = CD, these two distances are automatically equal.
Try it — Equal Chords Extended
  1. Two chords are both 12 cm long. If one is 5 cm from the centre, how far is the other from the centre?
  2. Two chords are 6 cm and 4 cm from the centre respectively. Can they be the same length?
Show answers
  1. 5 cm (equal chords are equidistant from the centre)
  2. No — if the chords were equal in length, they’d have to be the same distance from the centre; being different distances means they must be different lengths.

8. The Perpendicular Bisector of a Chord Passes Through the Centre Extended only

Using the same diagram as above: a line from the centre that meets a chord at a right angle always cuts that chord exactly in half.

Worked example A chord AB has length 16 cm. A perpendicular line is drawn from the centre O to the chord, meeting it at point M. Find AM and MB.
The perpendicular from the centre bisects the chord, so AM = MB = 16 ÷ 2 = 8 cm each
Worked example — using Pythagoras with this theorem A circle has radius 10 cm. A chord is 6 cm from the centre. Find the length of the chord.
The perpendicular from the centre bisects the chord, forming a right-angled triangle with the radius as the hypotenuse.
Half-chord = √(10² − 6²) = √(100−36) = √64 = 8 cm
Full chord length = 8 × 2 = 16 cm
Try it — Perpendicular Bisector of a Chord Extended
  1. A chord is 20 cm long. Find the distance from the midpoint to each end.
  2. A circle has radius 13 cm. A chord is 5 cm from the centre. Find the chord’s length.
  3. A circle has radius 15 cm and a chord of length 24 cm. Find the distance from the centre to the chord.
Show answers
  1. 10 cm each
  2. Half-chord = √(169−25) = √144 = 12. Full length = 24 cm
  3. Half-chord = 12. Distance = √(15²−12²) = √(225−144) = √81 = 9 cm

9. Tangents from an External Point Are Equal in Length Extended only

From any point outside a circle, exactly two tangent lines can be drawn to it — and they are always the same length.

Diagram and worked example O P T₁ T₂ P is a point outside the circle. Two tangents touch the circle at T₁ and T₂. This theorem guarantees PT₁ = PT₂, always — regardless of where P sits outside the circle.
Worked example — combining with Pythagoras A circle has radius 7 cm. An external point P is 25 cm from the centre. Find the length of each tangent from P.
The radius meets the tangent at 90° (Theorem 2), so OTP is a right-angled triangle with hypotenuse OP.
Tangent length = √(25² − 7²) = √(625−49) = √576 = 24 cm
Try it — Tangents from an External Point Extended
  1. Two tangents are drawn from an external point P. One has length 9 cm. Find the length of the other.
  2. A circle has radius 5 cm. An external point is 13 cm from the centre. Find the tangent length.
  3. A circle has radius 9 cm. A tangent from an external point has length 12 cm. Find the distance from the external point to the centre.
Show answers
  1. 9 cm (tangents from the same external point are equal)
  2. √(169−25) = √144 = 12 cm
  3. Distance = √(9²+12²) = √(81+144) = √225 = 15 cm
Common mistakes to watch for
  • Applying “angle at the centre is double” the wrong way round — always check which angle is at the centre (double) and which is at the circumference (half).
  • Assuming any two angles in a circle diagram are “in the same segment” without checking they’re subtended by the same chord from the same arc.
  • Forgetting cyclic quadrilateral angles are opposite pairs (A+C, B+D) — adjacent angles don’t necessarily sum to 180°.
  • Mixing up which side of the tangent/chord the alternate segment angle should be measured from — it’s always the angle on the far side of the chord from where the tangent angle was measured.
  • Forgetting the tangent-radius right angle is available as a tool even in problems that look like they’re about a completely different theorem (tangent length, alternate segment) — it often supplies the missing right-angled triangle.

10. Quick-Fire Challenge Round

True or False?
  1. The angle in a semicircle is always 90°.
  2. The angle at the centre is half the angle at the circumference.
  3. Opposite angles in a cyclic quadrilateral are always equal.
  4. Tangents drawn from the same external point are always equal in length.
Show answers
  1. True.
  2. False — it’s the other way round: the angle at the centre is double the angle at the circumference. Ext.
  3. False — opposite angles in a cyclic quadrilateral sum to 180°, they aren’t necessarily equal. Ext.
  4. True. Ext.
Spot the Error
  1. “Angle at the centre = 80°, so angle at the circumference = 80 × 2 = 160°.”
  2. “In a cyclic quadrilateral, angle A = 100° and angle B = 80°, so angle C = 180 − 100 = 80°.”
  3. “Two tangents from an external point are 8 cm and 10 cm — that’s fine, tangent lengths can differ.”
Show answers
  1. Angle at the centre is double the circumference angle, so the circumference angle is half the centre angle: 80÷2 = 40°, not 80×2.
  2. Angle C is opposite angle A, not angle B — this part is actually correct (180−100=80° for angle C), but it’s worth checking angle D too: 180−80=100° (opposite B). The reasoning shown for C is valid, but be careful to pair up the correct opposite angles.
  3. Two tangents from the same external point must always be equal — 8 cm and 10 cm would be a contradiction, meaning something in the problem must be inconsistent or these come from two different external points.

11. Mixed Practice — Bringing It All Together

Core & Extended
  1. AB is a diameter. C is on the circle, with angle BAC = 28°. Find angle ACB and angle ABC.
  2. A radius meets a tangent at T. What is the angle between them?
Show answers
  1. Angle ACB = 90° (semicircle). Angle ABC = 180−90−28 = 62°
  2. 90°
Extended-level mixed set Extended only
  1. Angle at the centre = 136°. Find the angle at the circumference.
  2. A cyclic quadrilateral has angle A = 78°. Find angle C.
  3. A tangent-chord angle is 41°. Find the angle in the alternate segment.
  4. A circle has radius 25 cm. A chord is 20 cm from the centre. Find the chord’s length.
Show answers
  1. 136÷2 = 68°
  2. 180−78 = 102°
  3. 41°
  4. Half-chord=√(625−400)=√225=15. Full length=30 cm

Practice at Home

A large set again, to close out the Geometry unit — take it steadily.

Part A — Semicircle and tangent-radius (Core & Extended)
  1. AB is a diameter, C is on the circle, angle ABC = 51°. Find angle ACB and angle BAC.
  2. A tangent and radius meet at T. A second angle at T (between the tangent and a chord) is 34°. Find the angle between the radius and the chord.
Part B — Centre/circumference and same segment Extended
  1. Angle at the centre = 88°. Find the angle at the circumference.
  2. Angle at the circumference = 55°. Find the angle at the centre.
  3. C and D are on the same arc, both subtending chord AB. Angle ACB = 63°. Find angle ADB.
Part C — Cyclic quadrilaterals and alternate segment Extended
  1. A cyclic quadrilateral has angle B = 114°. Find angle D.
  2. A cyclic quadrilateral has angles A=65°, B=x°, C=y°, D=100°. Find x and y.
  3. A tangent-chord angle is 72°. Find the angle in the alternate segment.
Part D — Chords and tangent lengths Extended
  1. Two equal chords are drawn in a circle. One is 7 cm from the centre. How far is the other from the centre?
  2. A circle has radius 17 cm. A chord is 8 cm from the centre. Find the chord’s length.
  3. Two tangents from an external point are drawn; one is 15 cm long. Find the other.
  4. A circle has radius 6 cm. A tangent from an external point is 8 cm long. Find the distance from the point to the centre.
Part E — Applied / challenge
  1. AB is a diameter of a circle. C is a point on the circle such that AC = BC. Find all three angles of triangle ABC, explaining your reasoning.
  2. A cyclic quadrilateral ABCD has angle A = (2x+10)° and angle C = (3x−30)°. Find x, then find angle A.
  3. A circle has centre O and radius 10 cm. Points A and B are on the circle such that chord AB = 12 cm. Find the perpendicular distance from O to AB, and use it (along with any other circle facts you need) to find angle AOB.

Key Vocabulary Recap

Subtend — an angle “subtended” by a chord/arc is the angle formed by lines drawn from that chord/arc to a point
Cyclic quadrilateral — a four-sided shape with all vertices on one circle Ext.
Segment — the region cut off by a chord
Alternate segment — the segment on the opposite side of a chord from a given angle Ext.
Equidistant — the same distance from

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) section C4.7 (Core subject content) and sections E4.7 and E4.8 (Extended subject content) — Circle theorems. Core covers only the angle-in-a-semicircle and tangent-radius theorems. Extended adds the angle at the centre/circumference relationship, angles in the same segment, cyclic quadrilaterals, the alternate segment theorem, equal chords, the perpendicular bisector of a chord, and tangents from an external point — all marked “Extended only” above. Core’s equivalent to E4.8 (section C4.8) is explicitly marked “Extended content only” in the syllabus. This is the final note in the Geometry unit.

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