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Circle Theorems

Revise Circle Theorems for Mathematics 0580 (O Level) — revision notes, 63 practice questions and instant AI marking.

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Circle Theorems

Cambridge IGCSE Extended Mathematics

🎯 The Big Idea: Circle theorems are rules about angles and lines in circles that let you find missing angles and lengths when you know others—they're all about spotting patterns in how things relate to the centre, tangents, and chords.

Quick Summary

Angles at Centre & Circumference

The angle at the centre is twice the angle at the circumference (same arc).

Angle in a Semicircle

Any angle inscribed in a semicircle is always 90°.

Chords & Perpendicular Bisector

A radius to a chord's midpoint meets it at right angles.

Radius & Tangent

A radius and a tangent meet at the point of contact at 90°.

Cyclic Quadrilaterals

Opposite angles in a cyclic quadrilateral add to 180°.

Angles in Same Segment

Angles from the same chord on the same side are equal.

Tangents from External Point

Two tangents from outside the circle are equal in length.

Alternate Segment Theorem

Angle between chord and tangent equals the angle in the opposite segment.

Angles at Centre & Circumference

What This Theorem Means

Imagine you draw two radii (lines from the centre to the edge) that meet the circle at the same two points. Now, from those same two points, draw a line to any other point on the circle's edge. The angle you make at the centre is always twice as big as the angle you make at the edge.

The key phrase you'll hear: both angles are "subtended by the same arc" — meaning they're both formed using the same two endpoints on the circle.

Circle Theorem
Angle at centre = 2 × Angle at circumference
(both subtended by the same arc)

How to Spot It on a Diagram

  1. Find two radii coming from the centre
  2. Follow them to the circumference — they touch at two points
  3. Look for a line from those two points to anywhere else on the circle's edge
  4. It often looks like an arrowhead shape
💡 Tip: This theorem still works even when the angles "overlap" or when they form a diamond shape. Just make sure you're comparing the right angles — the ones formed from the same two points on the circumference.
Worked Example 1

Question: A circle has centre O. Points A, B, and C are on the circumference. Angle AOB = 150°. Find angle ACB.

Step 1: Identify the theorem
We have an angle at the centre (AOB) and need to find an angle at the circumference (ACB). Both are subtended by the same arc AB.
The angles at centre & circumference theorem applies.
Step 2: Apply the theorem
The angle at the centre is twice the angle at the circumference.
Angle AOB = 2 × Angle ACB
Step 3: Substitute and solve
150 = 2 × Angle ACB
Angle ACB = 150 ÷ 2 = 75°
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What to Memorise

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Also in the full note
  • Theorems with Chords & Tangents
  • Angles in Cyclic Quadrilaterals
  • Angles in the Same Segment
  • The Alternate Segment Theorem
  • Concepts Checklist
  • Exam Tips & Common Pitfalls
  • Ready to Revise?
  • Common Mistakes to Avoid
What's inside
📖 Revision notes ◉ 63 practice questions 🎯 Learn mode ✦ AI flashcards ✓ Instant AI marking 🧊 3D explorers 🧪 Experiments & simulations 📈 Progress tracking

Past papers that test Circle Theorems

Real Mathematics 0580 papers with questions on this topic — open one and get it marked instantly, free.

0580 Nov 2024 · Paper 3 · Variant 3 Cambridge · mark scheme Open → 0580 Jun 2021 · Paper 3 · Variant 1 Cambridge · mark scheme Open → 0580 Nov 2019 · Paper 3 · Variant 2 Cambridge · mark scheme Open → 0580 Nov 2018 · Paper 2 · Variant 2 Cambridge · mark scheme Open → 0580 Jun 2018 · Paper 4 · Variant 1 Cambridge · mark scheme Open → 0580 Nov 2017 · Paper 3 · Variant 2 Cambridge · mark scheme Open →
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