What are the basic Circle Theorems covered in the Cambridge IGCSE Mathematics curriculum?

The Cambridge IGCSE Mathematics curriculum covers several key Circle Theorems, including the Angle at the Centre Theorem, the Angle in a Semicircle Theorem, the Angles in the Same Segment Theorem, the Cyclic Quadrilateral Theorem, and the Tangent-Secant Theorem. These theorems help in understanding the properties and relationships between angles and segments in and around circles.

How does the Angle at the Centre Theorem work in circle geometry?

The Angle at the Centre Theorem states that the angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the circumference. This theorem is fundamental in solving problems related to angles in circle geometry, as it establishes a clear relationship between central and inscribed angles.

Can you explain the Cyclic Quadrilateral Theorem?

The Cyclic Quadrilateral Theorem states that in a cyclic quadrilateral (a four-sided figure where all vertices lie on the circumference of a circle), the opposite angles sum up to 180 degrees. This theorem is crucial for solving problems involving cyclic quadrilaterals and is often used in conjunction with other circle theorems to find unknown angles.

What is the significance of the Tangent-Secant Theorem in circle geometry?

The Tangent-Secant Theorem, also known as the Power of a Point Theorem, states that if a tangent and a secant are drawn from a point outside a circle, then the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external segment. This theorem is significant for solving problems involving lengths and distances in circle geometry.

How can the Angle in a Semicircle Theorem be applied in solving geometry problems?

The Angle in a Semicircle Theorem states that any angle subtended by a diameter on the circumference of a circle is a right angle (90 degrees). This theorem is particularly useful in solving problems where right angles need to be identified or verified, and it is often applied in conjunction with other geometric principles to solve complex problems in circle geometry.

Why is "Stating the angle without naming the theorem" a common mistake in Circle Theorems?

Why it happens: Students see the answer and write it. How to avoid it: Mark schemes credit the THEOREM name — "angle at centre", "same segment", "alternate segment" etc. Always include it. Source: 0580/42 — every series

Why is "Doubling instead of halving (or vice versa) for centre/circumference" a common mistake in Circle Theorems?

Why it happens: Students forget which is twice the other. How to avoid it: Mnemonic: angle at the CENTRE is BIGGER. So centre = 2 × circumference.

Why is "Pairing adjacent angles in a cyclic quadrilateral" a common mistake in Circle Theorems?

Why it happens: Confusing "opposite" with "adjacent". How to avoid it: Opposite angles in a cyclic quad sum to 180 — they're across from each other, not next to each other.

Why is "Picking an angle from the wrong segment" a common mistake in Circle Theorems?

Why it happens: Confusing which segment is "alternate". How to avoid it: Alternate segment is on the OTHER side of the chord from the tangent angle.

GeometryCambridge IGCSE Mathematics (0580)

Circle Theorems

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next.

Previous Next

Learn this Topic

Start with the slides for the quick version, then go deeper with the full study notes.

Short Study Notes in the form of Slides

Read the notes first. If the method in a worked example clicks, you're ready for the questions.

Page 1 / 0

Detailed Study Notes

Full prose, callouts and a recap — built for A* mastery, not just a quick scan.

Take these study notes with you

Download a branded PDF — full prose, callouts, recap and memorise list for Circle Theorems, ready to print or save offline.

Circle Theorems — Cambridge IGCSE 0580 Maths Extended (2026)

The seven named theorems that turn diagrams into angle equations. Cambridge mark schemes credit the theorem name as much as the answer — memorise them precisely.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E4.7 — Use the following circle theorems: angle at centre, angles in same segment, angle in semicircle, cyclic quadrilateral, tangent-radius, equal tangents, alternate segment.

Angle at centre = 2 × angle at circumference

Same arc, same side. The centre angle is double.

If two angles in a circle subtend the same arc — one at the centre and one at the circumference (on the same side) — the angle at the centre is twice the angle at the circumference.

Worked. Centre angle = $130\degree$ . Find the circumference angle subtending the same arc.

$\theta = 130 / 2 = 65\degree$ .
Reason: angle at centre is twice angle at circumference.

Special case. If the chord is a DIAMETER, the centre angle is $180\degree$ , so the circumference angle is $90\degree$ — that's the angle in a semicircle = $90\degree$ theorem.

Same arc, same side.
Centre angle = $2 \times$ circumference angle.
Diameter case: angle in semicircle = $90\degree$ .
ALWAYS state the theorem name in the reason.

Angles in the same segment

All angles subtended by the same chord, on the same side, are equal.

Theorem. If multiple angles at the circumference are subtended by the SAME chord, on the SAME side of that chord, they are all EQUAL.

Worked. Two angles at the circumference subtended by the same chord, on the same side. One is $48\degree$ . The other:

$48\degree$ .
Reason: angles in the same segment are equal.

Tip. Look for two angles whose vertices are on the same arc and whose sides reach the same two endpoints (the chord).

Same chord + same side → equal angles.
Cite the theorem by name.

Cyclic quadrilateral

Opposite angles in a cyclic quadrilateral sum to $180\degree$ .

A cyclic quadrilateral has all four vertices on the circle.

Theorem. Opposite angles of a cyclic quadrilateral sum to $180\degree$ .

Worked. $ABCD$ is cyclic. $\angle A = 78\degree$ . Find $\angle C$ .

$\angle A + \angle C = 180\degree \Rightarrow \angle C = 102\degree$ .
Reason: opposite angles in a cyclic quadrilateral.

Don't confuse with adjacent. Opposite means ACROSS, not next to. Adjacent angles in a cyclic quad don't generally satisfy the rule.

All four vertices on the circle.
Opposite angles sum to $180\degree$ .
Two pairs of opposite angles each sum to $180\degree$ .

Tangent properties

Tangent ⊥ radius at the point of contact. Two tangents from an external point are equal.

Tangent perpendicular to radius. A tangent meets the radius drawn to the point of contact at $90\degree$ .

Worked. Tangent $TA$ touches circle at $A$ ; centre $O$ . The angle $\angle OAT = 90\degree$ .

Reason: tangent perpendicular to radius at point of contact.

Two tangents from an external point. If two tangents are drawn to a circle from an external point $P$ , the two tangent lengths are EQUAL.

Worked. $PA$ and $PB$ are tangents to a circle from $P$ , touching at $A$ and $B$ . Then $PA = PB$ .

Reason: tangents from an external point are equal.

Alternate segment theorem. The angle between a tangent and a chord at the point of contact equals the angle in the alternate (opposite) segment subtended by that chord.

Worked. Tangent $TA$ at $A$ . Chord $AB$ makes angle $42\degree$ with the tangent. Find the angle in the alternate segment subtended by $AB$ .

$42\degree$ .
Reason: alternate segment theorem.

Tangent ⊥ radius at the contact point.
Two tangents from external point: equal lengths.
Alternate segment: angle between tangent and chord = angle in opposite segment.
Cite each theorem by NAME for full marks.

How it’s examined

Circle theorems are guaranteed on Paper 4 (5-8 marks). Cambridge stacks 2-3 theorems in one diagram and asks for an angle, with reasons. Paper 2 has simpler 2-3 mark single-theorem items. Examiner reports flag failure to STATE the theorem name as the recurring lost-mark scenario — even when the angle calculation is correct.

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E4.7); 0580/42 Oct/Nov 2024 — Q9 (cyclic quadrilateral); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

Master this Topic

Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

Take this whole topic with you

Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Circle Theorems, ready to print or save as PDF.

Step-by-step worked examples — Circle Theorems

Step-by-step solutions to past-paper-style questions on circle theorems, written exactly the way a tutor would explain them at the board.

1Use "angle at the centre is twice angle at the circumference"
Extended• centre theorem
Question
$O$ is the centre. Angle at the centre subtended by arc $AB$ is $130\degree$ . Find the angle at the circumference subtended by the same arc.
Step-by-step solution
1. Step 1
  Angle at centre = $2\times$ angle at circumference (same arc, same side).
  $\theta = \dfrac{130}{2} = 65\degree$
Answer
$65\degree$
2Cyclic quadrilateral
Extended• Adapted from 0580/42 May/Jun 2024 Q9• cyclic quadrilateral
Question
$ABCD$ is a cyclic quadrilateral. $\angle A = 78\degree$ and $\angle B = 102\degree$ . Find $\angle C$ and $\angle D$ .
Step-by-step solution
1. Step 1
  Opposite angles in a cyclic quadrilateral sum to $180\degree$ .
  $\angle C = 180 - 78 = 102\degree$
2. Step 2
  Same rule for $\angle D$ .
  $\angle D = 180 - 102 = 78\degree$
Answer
$\angle C = 102\degree,\ \angle D = 78\degree$
3Tangent perpendicular to radius
Extended• tangent
Question
A tangent touches a circle at $T$ . $OT$ is the radius. The angle $\angle OTQ$ (between tangent and radius) is given. State its value.
Step-by-step solution
1. Step 1
  The tangent at any point is perpendicular to the radius drawn to that point.
Answer
$90\degree$
4Use the alternate segment theorem
Extended• alternate segment
Question
Line $TA$ is tangent at $A$ . Chord $AB$ creates an angle of $42\degree$ with the tangent. Find the angle in the alternate segment subtended by $AB$ .
Step-by-step solution
1. Step 1
  Alternate segment: angle between tangent and chord = angle in the alternate segment.
Answer
$42\degree$
Examiner tip
Always cite "alternate segment theorem" by name — mark schemes credit it explicitly.
5Angles in the same segment
Extended• same segment
Question
Two angles are subtended at the circumference by the same chord, on the same side. One is $48\degree$ . Find the other.
Step-by-step solution
1. Step 1
  Angles in the same segment are equal.
Answer
$48\degree$

Key Formulae — Circle Theorems

The formulae you need to memorise for circle theorems on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Angle at centre = 2 × angle at circumference
$\theta_{\text{centre}} = 2 \times \theta_{\text{circumference}}$
When to use
When two angles subtend the same arc, one at the centre and one on the circumference (same side).
Angles in the same segment are equal
$\theta_{1} = \theta_{2}$
When to use
Two angles at the circumference subtending the same chord, on the same side.
Cyclic quadrilateral
$\angle A + \angle C = 180\degree,\quad \angle B + \angle D = 180\degree$
When to use
Opposite angles in a cyclic quadrilateral sum to $180\degree$ .
Tangent ⊥ radius
$\angle (OT, \text{tangent}) = 90\degree$
When to use
Tangent meets the radius drawn to the point of contact at $90\degree$ .
Alternate segment
$\angle (\text{tangent, chord}) = \angle (\text{in alternate segment})$
When to use
Angle between tangent and chord at the point of contact equals the inscribed angle in the alternate segment.

Key Definitions and Keywords — Circle Theorems

Definitions to memorise and the exact keywords mark schemes credit for circle theorems answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Tangent
Examiner keyword
A line that touches a circle at exactly one point.
Chord
Examiner keyword
A straight line joining two points on a circle.
Cyclic quadrilateral
Examiner keyword
A quadrilateral whose four vertices lie on a single circle.
Segment of a circle
The region bounded by a chord and an arc.
Inscribed (circumference) angle
An angle whose vertex is on the circumference and whose two sides are chords.

Common Mistakes and Misconceptions — Circle Theorems

The traps other students keep falling into on circle theorems questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Stating the angle without naming the theorem
0580/42 — every series
Why it happens
Students see the answer and write it.
How to avoid it
Mark schemes credit the THEOREM name — "angle at centre", "same segment", "alternate segment" etc. Always include it.
✕Doubling instead of halving (or vice versa) for centre/circumference
Why it happens
Students forget which is twice the other.
How to avoid it
Mnemonic: angle at the CENTRE is BIGGER. So centre = 2 × circumference.
✕Pairing adjacent angles in a cyclic quadrilateral
Why it happens
Confusing "opposite" with "adjacent".
How to avoid it
Opposite angles in a cyclic quad sum to $180\degree$ — they're across from each other, not next to each other.
✕Picking an angle from the wrong segment
Why it happens
Confusing which segment is "alternate".
How to avoid it
Alternate segment is on the OTHER side of the chord from the tangent angle.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Video lesson

Short walkthrough of the concepts students most often get stuck on.

Circle Theorems — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Circle Theorems

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Use "angle at the centre is twice angle at the circumference"

2Cyclic quadrilateral

3Tangent perpendicular to radius

4Use the alternate segment theorem

5Angles in the same segment

Angle at centre = 2 × angle at circumference

Angles in the same segment are equal

Cyclic quadrilateral

Tangent ⊥ radius

Alternate segment

Tangent

Chord

Cyclic quadrilateral

Segment of a circle

Inscribed (circumference) angle

✕Stating the angle without naming the theorem

✕Doubling instead of halving (or vice versa) for centre/circumference

✕Pairing adjacent angles in a cyclic quadrilateral

✕Picking an angle from the wrong segment

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Circle Theorems — frequently asked questions