Circle Theorems
Circle theorem questions are unusual in IGCSE Maths: the reason is worth as much as the number. You will be asked to “give a reason for your answer”, and a correct angle with no justification routinely scores half. Learn the theorems with their names.
1. The vocabulary first
| Term | Meaning |
|---|---|
| Radius | Centre to the edge |
| Diameter | Right across through the centre — twice the radius |
| Chord | A line joining two points on the circle (not through the centre) |
| Tangent | A line touching the circle at exactly one point |
| Arc | Part of the circumference |
| Sector | A “pizza slice” — bounded by two radii and an arc |
| Segment | Bounded by a chord and an arc |
| Cyclic quadrilateral | A quadrilateral with all four vertices on the circle |
| Subtend | To “open out” an angle from — an arc subtends an angle at a point |
A tangent touches at one point only; it does not cross the circle. And a segment is not a sector — a sector is cut by two radii, a segment by a chord.
2. The seven theorems
1. Angle in a semicircle is 90°
The angle subtended by a diameter at the circumference is 90°.
Reason to write: “Angle in a semicircle is 90°.”
Only if the line really is a diameter — it must pass through the centre. Assuming a right angle when the chord is not a diameter is a documented error. Conversely, if you’re told the angle is 90°, you can conclude the line is a diameter.
2. Angle at the centre is twice the angle at the circumference
The angle at the centre is double the angle at the circumference, when both are subtended by the same arc.
Reason: “Angle at centre is twice the angle at circumference.”
Both angles must stand on the same arc. Check the two angles “point at” the same two endpoints before doubling or halving.
Which way round? Centre = 2 × circumference. If you know the one at the circumference, double it; if you know the one at the centre, halve it.
3. Angles in the same segment are equal
Angles subtended by the same arc, at the circumference, are equal.
Reason: “Angles in the same segment are equal.”
These appear as a “bow tie” or arrowhead shape. Both angles must be on the same side of the chord.
4. Opposite angles of a cyclic quadrilateral add to 180°
In a cyclic quadrilateral, opposite angles are supplementary — they sum to 180°.
Reason: “Opposite angles in a cyclic quadrilateral add up to 180°.”
This is only true for CYCLIC quadrilaterals — all four corners on the circle. Opposite angles of a general quadrilateral have no such property. Applying it to any four-sided shape was a repeated error.
Adding them to 90° instead of 180° is another frequent slip. Also useful: the exterior angle of a cyclic quadrilateral equals the interior opposite angle.
5. A tangent meets a radius at 90°
The angle between a tangent and the radius at the point of contact is 90°.
Reason: “The angle between a tangent and radius is 90°.”
This right angle is often not marked on the diagram. You are expected to know it — and it is usually the key that unlocks the whole question. Whenever you see a tangent, mark the right angle in yourself.
6. Two tangents from a point are equal
Tangents drawn to a circle from the same external point are equal in length.
Reason: “Tangents from an external point are equal.”
This creates an isosceles triangle, which then gives you a pair of equal base angles — often the next step in the question.
7. Alternate segment theorem
The angle between a tangent and a chord equals the angle in the alternate segment.
Reason: “Alternate segment theorem.”
This is the theorem students know least well — it came up as a difficulty far more often than any other. The angle between the tangent and the chord is equal to the angle subtended by that chord in the segment on the other side.
How to spot it: you have a tangent, and a chord leaving the point of contact. Follow the chord across the circle; the angle at the far vertex, on the opposite side of the chord, is the equal one.
3. Two more facts that finish most questions
The perpendicular from the centre to a chord bisects it.
A line from the centre meeting a chord at 90° cuts it exactly in half — and the reverse also holds. This gives you a right-angled triangle, so Pythagoras usually follows.
Any triangle with two radii is isosceles.
All radii are equal, so a triangle formed by two radii has two equal sides and therefore two equal base angles. This is the most-used unstated step in circle geometry.
But don’t assume isosceles without a reason. Marking a triangle isosceles when its two equal sides aren’t radii — and aren’t given — was a recorded error. State why: “OA = OB (radii)“.
4. Method — how to attack a circle question
- Mark everything you know on the diagram — the right angle at any tangent, equal radii, given angles
- Look for the trigger words: diameter → semicircle theorem; tangent → 90° or alternate segment; four points on the circle → cyclic quadrilateral
- Work in small steps, writing the reason for each
- Use ordinary angle facts too — angles in a triangle sum to 180°, angles on a straight line 180°, angles round a point 360°, and alternate/corresponding angles if parallel lines appear
- Check the total makes sense
Write a reason at every step, naming the theorem. For a 3- or 4-mark question, the marks are typically split between the working and the reasons. Tutors made this point repeatedly — an unexplained correct answer is an incomplete answer.
Never assume from the picture. Diagrams are not to scale. An angle that looks like 90° is 90° only if it is marked, given, or follows from a theorem.
5. Scope note — what is not on 0580
Some lessons in this area covered material from Additional Mathematics (0606) rather than 0580:
- the equation of a circle, (x − a)² + (y − b)² = r²
- finding the equation of a tangent to a circle
- using the discriminant b² − 4ac to prove a line is a tangent
If you are sitting 0580 only, these are not on your syllabus and you can safely skip them. They are genuinely useful for 0606 or A-level, but they are not examinable on 0580 — check your own syllabus document before spending revision time here.
6. Mistakes that cost marks
Giving the angle without a reason.
Applying cyclic quadrilateral rules to a non-cyclic quadrilateral.
Using 90° instead of 180° for opposite angles in a cyclic quadrilateral.
Assuming a chord is a diameter to claim a right angle.
Missing the tangent–radius right angle because it isn’t marked.
Doubling instead of halving between centre and circumference — or using angles on different arcs.
Not recognising the alternate segment theorem.
Assuming a triangle is isosceles without justifying it with radii.
Measuring the diagram rather than reasoning.
Confusing a sector with a segment.
Frequently asked questions
What is the angle in a semicircle? 90° — the angle subtended by a diameter at the circumference.
What is the relationship between the angle at the centre and at the circumference? The angle at the centre is twice the angle at the circumference, when both stand on the same arc.
What are the angle rules for a cyclic quadrilateral? Opposite angles add to 180°, and the exterior angle equals the interior opposite angle. All four vertices must be on the circle.
What angle does a tangent make with a radius? 90°, at the point of contact — even when the diagram doesn’t mark it.
What is the alternate segment theorem? The angle between a tangent and a chord equals the angle in the alternate segment — the angle subtended by that chord on the other side.
Are two tangents from the same point equal? Yes, which makes an isosceles triangle.
What happens when a line from the centre meets a chord at right angles? It bisects the chord.
Do I have to write reasons? Yes. Reasons carry marks, and naming the theorem is the safest way to earn them.
Is the equation of a circle on 0580? No — that belongs to Additional Mathematics 0606.
Quick revision checklist
- I know the vocabulary, including segment vs sector
- Angle in a semicircle is 90° — and I check it really is a diameter
- Angle at centre is twice angle at circumference, on the same arc
- Angles in the same segment are equal
- Opposite angles of a cyclic quadrilateral add to 180°
- I know the exterior angle rule for cyclic quadrilaterals
- Tangent meets radius at 90°, marked or not
- Tangents from an external point are equal, giving an isosceles triangle
- I can recognise and apply the alternate segment theorem
- Perpendicular from the centre bisects a chord
- I justify isosceles triangles with “radii”
- I write a named reason at every step
- I never assume angles from how the diagram looks
- I know which circle material is 0606 and not 0580
These notes cover circle theorems in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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