IGCSE Circle Theorems: Complete Guide for Cambridge IGCSE Mathematics

IGCSE circle theorems are essential geometry topics in Cambridge IGCSE Mathematics that appear in both Paper 2 and Paper 4. Mastering all circle theorems, angle at center, angle in semicircle, and cyclic quadrilaterals is essential for solving circle geometry problems.

This comprehensive IGCSE circle theorems guide covers everything you need to know, including all 8 circle theorems, worked examples, common exam questions, and expert tips from Tutopiya’s IGCSE maths tutors. We’ll also show you how to avoid the most common mistakes that cost students valuable marks.

🎯 What you’ll learn: By the end of this guide, you’ll know all circle theorems, how to apply them to find missing angles, and apply these skills to solve problems in IGCSE exams.

Already studying with Tutopiya? Practice these skills with our dedicated IGCSE Geometry practice deck featuring exam-style questions and instant feedback.

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Why IGCSE Circle Theorems Matter

IGCSE circle theorems are essential geometry topics. Here’s why they’re so important:

• High frequency topic: Circle theorem questions appear regularly in IGCSE maths papers
• Foundation skill: Essential for advanced circle geometry
• Exam weight: Typically worth 6-10 marks per paper
• Real-world applications: Used in design, engineering, and construction
• Problem-solving skills: Develops geometric reasoning and theorem application

Key insight from examiners: Students often struggle to identify which theorem to use or make errors with angle calculations. This guide will help you master all theorems systematically.

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Understanding Circle Theorems

Circle theorems are rules about angles and lines in circles.

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Theorem 1: Angle at Center

Angle at center = 2 × angle at circumference (angles subtended by same arc)

Example 1: Angle at center is 80°. Find angle at circumference.

Solution: Angle at circumference = 80° / 2 = 40°

Answer: 40°

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Theorem 2: Angle in Semicircle

Angle in semicircle = 90°

Example 2: In a semicircle, the angle at the circumference is always 90°.

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Theorem 3: Angles in Same Segment

Angles in same segment are equal

Example 3: Two angles subtended by the same arc are equal.

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Theorem 4: Cyclic Quadrilateral

Opposite angles in cyclic quadrilateral add to 180°

Example 4: In a cyclic quadrilateral, one angle is 70°. Find the opposite angle.

Solution: Opposite angle = 180° - 70° = 110°

Answer: 110°

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Theorem 5: Tangent and Radius

Tangent is perpendicular to radius at point of contact

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Theorem 6: Alternate Segment Theorem

Angle between tangent and chord = angle in alternate segment

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Theorem 7: Tangents from External Point

Tangents from external point are equal in length

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Theorem 8: Angle Between Tangent and Chord

Angle between tangent and chord = angle in opposite segment

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Common Examiner Traps

• Theorem identification - Learn to recognize which theorem applies
• Angle calculation errors - Double-check arithmetic
• Diagram interpretation - Label angles clearly on diagrams

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Practice Questions

Question 1

In a circle, angle at center is 120°. Find angle at circumference.

Solution: Angle at circumference = 120° / 2 = 60°

Answer: 60°

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Frequently Asked Questions About IGCSE Circle Theorems

What is the angle at center theorem?

Angle at center = 2 × angle at circumference when both angles are subtended by the same arc.

What is the angle in semicircle theorem?

Angle in semicircle = 90° - any angle in a semicircle is a right angle.

What is a cyclic quadrilateral?

A cyclic quadrilateral is a four-sided shape with all vertices on a circle. Opposite angles add to 180°.

What is the alternate segment theorem?

Alternate segment theorem: Angle between tangent and chord equals angle in the alternate segment.

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Related IGCSE Maths Resources

Strengthen your IGCSE Mathematics preparation with these comprehensive guides:

• IGCSE Angle Theorems: Complete Guide - Master angle theorems
• IGCSE Circles: Complete Guide - Master circle properties
• IGCSE Maths Revision Notes, Syllabus and Preparation Tips - Complete syllabus overview, topic breakdown, and revision strategies
• IGCSE Past Papers Guide - Access free IGCSE past papers and exam resources

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Next Steps: Master IGCSE Circle Theorems with Tutopiya

Ready to excel in IGCSE circle theorems? Our expert IGCSE maths tutors provide:

• Personalized 1-on-1 tutoring tailored to your learning pace
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Book a free IGCSE maths trial lesson and get personalized support to master circle theorems and achieve your target grade.

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