Circle Theorems
CIE IGCSE Mathematics· 4.7, E4.8· 25 min read
1. Core Required Circle Theorems★★☆☆☆⏱ 10 min
Core tier students only need to memorize and apply two circle theorems for all Core paper questions. No other circle rules are required for Core assessments.
Angle in a Semicircle
The angle subtended at the circumference of a circle by a diameter is always a right angle (90°).
Example:
If AB is the diameter of a circle, and point C lies on the circumference, angle ACB = 90°.
Tangent Perpendicular to Radius
A tangent to a circle is always perpendicular to the radius at the point of contact.
Example:
If line T is tangent to a circle at point P, and OP is the radius, angle OPT = 90°.
AB is the diameter of a circle with centre O. Point C lies on the circumference. Angle CAB = 32°. Calculate the size of angle ABC, stating your reason.
- 1
Step 1: Apply the angle in a semicircle rule to find angle ACB:
- 2
Step 2: Use the sum of angles in a triangle (180°) to calculate angle ABC:
- 3
Step 3: Write your exam justification: Angle in a semicircle is a right angle.
Exam tip:
Core students must always state the exact syllabus reason for every angle calculation to earn full marks; marks are deducted for missing justifications.
2. Extended Additional Theorems (Angle Rules)★★★☆☆Extended only⏱ 8 min
Extended tier students must master all Core theorems plus four additional angle-related rules, listed below:
Angle at the centre of a circle is twice the angle at the circumference subtended by the same arc
Angles in the same segment of a circle, subtended by the same arc, are equal
Opposite angles of a cyclic quadrilateral sum to 180°
Alternate segment theorem: the angle between a tangent and a chord at the point of contact equals the angle in the alternate segment
O is the centre of a circle. Points A, B, and C lie on the circumference. Angle AOB = 110°. Calculate the size of angle ACB, stating your reason.
- 1
Step 1: Apply the angle at centre rule: angle at circumference = ½ × angle at centre
- 2
Step 2: Write your exam justification: Angle at the centre is twice the angle at the circumference.
3. Extended Additional Theorems (Tangent & Chord Rules)★★★★☆Extended only⏱ 7 min
Extended students also need to know three chord and tangent length rules, frequently tested in 2-3 mark questions:
Equal length chords are equidistant from the centre of the circle
The perpendicular bisector of a chord passes through the centre of the circle
Two tangents drawn to a circle from the same external point are equal in length
Two tangents are drawn from external point P to a circle, touching the circle at points A and B. PA = 7 cm. Calculate the length of PB, stating your reason.
- 1
Step 1: Apply the equal tangents rule: tangents from a single external point are equal
- 2
Step 2: Write your exam justification: Tangents from a common external point are equal in length.
4. Common Pitfalls
Wrong move:
Forgetting to state the theorem reason in answers
Why:
CIE awards explicit marks for correct justifications; missing them costs 1-2 marks per question even if the numerical answer is correct
Correct move:
Write the exact syllabus theorem name (e.g. "angle in a semicircle is a right angle") alongside every calculation step
Wrong move:
Reversing the angle at centre ratio (writing circumference angle = 2 × centre angle)
Why:
The centre angle is always larger for the same arc, so reversing the ratio gives a value double the correct answer
Correct move:
Check which angle is bigger in the diagram first, or remember the mnemonic: "Centre is double the circumference"
Wrong move:
Applying the alternate segment theorem to non-tangent lines
Why:
The rule only applies when one line is explicitly marked as a tangent to the circle
Correct move:
Confirm the line is labeled as a tangent before using the alternate segment theorem
Wrong move:
Using Extended theorems for Core tier questions
Why:
Core questions are designed to only require the two Core rules; using Extended rules increases the risk of misapplication and wastes time
Correct move:
Core students only use the semicircle and tangent perpendicular to radius rules for all Core paper questions
Wrong move:
Assuming all quadrilaterals with vertices on a circle are cyclic
Why:
While this is technically true, you must only apply cyclic quadrilateral rules if the question confirms the quadrilateral is cyclic or all vertices are shown on the circle
Correct move:
Verify all four vertices lie on the circle before using opposite angles sum to 180°
5. Quick Reference Cheatsheet
Theorem Name | Core/Extended | Rule | Exam Reason Phrase |
|---|---|---|---|
Angle in Semicircle | Core | Angle subtended by diameter at circumference = 90° | Angle in a semicircle is a right angle |
Tangent ⊥ Radius | Core | Tangent is perpendicular to radius at point of contact | Tangent is perpendicular to the radius |
Angle at Centre | Extended | Angle at centre = 2 × angle at circumference for same arc | Angle at centre is twice the angle at circumference |
Angles in Same Segment | Extended | Angles subtended by same arc in same segment are equal | Angles in the same segment are equal |
Cyclic Quadrilateral | Extended | Opposite angles sum to 180° | Opposite angles of a cyclic quadrilateral sum to 180° |
Alternate Segment Theorem | Extended | Angle between tangent and chord = angle in alternate segment | Alternate segment theorem |
Equal Chords Equidistant | Extended | Equal length chords are equidistant from centre | Equal chords are equidistant from the centre |
Equal Tangents | Extended | Two tangents from same external point are equal length | Tangents from a common external point are equal |
What's Next
Now that you have mastered all required circle theorems for CIE IGCSE Mathematics 0580, you are ready to tackle mixed geometry problems that combine these rules with other angle properties, congruent triangles, and trigonometry. Practice writing full justifications for every step, as this is the most common area where students lose marks in exam questions. For Core students, you can move straight to past paper circle theorem questions limited to the two Core rules. For Extended students, practice cyclic quadrilateral and alternate segment theorem problems first, as these are the most frequently tested extended topics. Make sure you can identify which theorem applies to a given diagram quickly to save time in exams.