Circle theorems are essential geometric rules that help you solve...
Maths164 views·Updated 2 Sept 2026·2 pagesThe 7 Circle Theorems Explained with Examples
1
of 2
Basic Circle Theorems
Radius and tangent relationships form the foundation of circle geometry. When a radius meets a tangent at the point where it touches the circle, they always form a perfect 90° angle - this is one of the most reliable rules you'll use.
Here's something quite clever about tangents: from any point outside a circle, you can draw exactly two tangent lines. The distances from that external point to where each tangent touches the circle are always equal. This equal tangent lengths rule often helps you find missing measurements.
The angle at the centre theorem shows a beautiful relationship between angles. Any angle formed at the centre of a circle is exactly twice the size of the angle formed at the circumference when both angles use the same arc. This 2:1 ratio appears constantly in circle problems.
Quick tip: Remember "centre doubles circumference" - the angle at the centre is always double the angle at the circumference for the same arc.
Finally, any angle in a semicircle is always 90°. If you spot a triangle where one side is the diameter of a circle, you've found yourself a right-angled triangle.
2
of 2
Advanced Circle Theorems
Angles in the same segment are always equal to each other. This means if you have multiple angles that sit on the same side of a chord and use the same arc, they'll all have identical measurements - quite handy for solving complex problems.
Cyclic quadrilaterals (four-sided shapes inscribed in circles) follow a special rule: opposite angles always add up to 180°. If you know three angles in a cyclic quadrilateral, you can instantly find the fourth one.
The alternate segment theorem is perhaps the trickiest but most elegant rule. The angle between a tangent and a chord equals the angle in the alternate segment. Think of it as angles "on opposite sides" being equal.
Exam strategy: These theorems often combine in questions, so look for multiple rules working together rather than just one isolated theorem.
Understanding these patterns means you'll recognise the theorems quickly in exam questions. Practice identifying which theorem applies first, then apply the rule confidently.
We thought you’d never ask...
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar content
Algebra 1Most popular content in Maths
9MathsComprehensive Maths Concepts
Explore essential mathematical concepts including powers, geometry, statistics, and probability. This resource features 65 pages of detailed explanations, diagrams, and examples to enhance your understanding of topics such as right triangles, volume calculations, and data representation. Ideal for students seeking to strengthen their numeracy skills and grasp complex mathematical principles.
1180,3536,327
MathsGCSE Maths (Higher) // Revision Guide
The only GCSE maths (higher) revision guide you need to get a grade 9! Contains every topic, each with all potential question types and their solutions.
112,71261
MathsComprehensive Maths Concepts
Explore essential mathematical concepts including polynomial theorems, logarithmic properties, trigonometric functions, and integration techniques. This resource covers everything from solving inequalities to understanding exponential functions, providing a solid foundation for A-level mathematics. Ideal for students aiming for top grades.
1222,0931,822
MathsFoundation Maths Exam Solutions
Explore detailed solutions for the Foundation Tier Non-Calculator Maths exam. This resource covers key concepts such as probability, volume calculations, data representation, and more. Perfect for students preparing for their GCSE Maths exam, with step-by-step explanations and examples.
1021,3821,243
MathsTrigonometric Functions Overview
Explore the fundamentals of trigonometry, including the tangent, sine, and cosine functions. This summary covers key concepts such as SOH CAH TOA, trigonometric ratios, and methods for finding angles and sides in right triangles. Ideal for students preparing for exams or needing a quick reference.
111,77559
MathsGCSE Maths 2018 Exam Insights
Explore the key concepts from the 2018 GCSE Maths Paper 2, including compound interest, probability, standard form, and geometric transformations. This comprehensive summary covers essential topics such as interest rates, area calculations, and Venn diagrams, providing students with a clear understanding of the exam's requirements. Ideal for exam preparation and practice.
98,229411
MathsGCSE Maths Foundation Checklist
Comprehensive revision checklist covering essential topics for the GCSE Maths Foundation tier, including statistics, geometry, algebra, probability, and trigonometry. Perfect for students aiming to pass their exams with confidence.
113,88752
MathsYear 8 Maths AQA Exam
Explore the AQA Year 8 Term 3 Main Paper 2, featuring comprehensive solutions to key mathematical concepts including geometry, percentages, sequences, and data representation. This resource covers essential topics such as area calculations, properties of shapes, and survey analysis, making it ideal for exam preparation and revision.
83,576138
MathsPythagorean Theorem Explained
Explore the Pythagorean Theorem (a² + b² = c²) with clear examples and step-by-step calculations. This summary covers key concepts such as hypotenuse and right triangles, making it essential for understanding geometry. Ideal for students preparing for exams or needing a quick reference.
115775
Most popular content
9
SociologySociology of Families: Comprehensive Revision
Dive into an extensive overview of family dynamics, perspectives, and patterns in sociology. This resource covers key concepts such as family diversity, gender roles, marriage, and the impact of social policies on family structures. Perfect for A-Level Sociology students preparing for Paper 2.
1274,0622,307
SociologySociology of Education Overview
Explore comprehensive A-Level Sociology notes on the education system, covering key theories, policies, and sociological perspectives. This resource includes insights on marketisation, gender roles, cultural deprivation, and educational inequalities, providing a thorough understanding of how education shapes social stratification and individual achievement. Ideal for exam preparation and in-depth study.
12103,2233,044
BiologyA-Level Biology Year 1 Overview
Comprehensive summary of AQA A-Level Biology Year 1, covering key topics such as cellular structure, protein synthesis, immune response, gas exchange, and more. Ideal for exam preparation and understanding biological concepts. Includes detailed insights into cellular processes, biological classification, and the circulatory system.
1215,192704
BiologyAQA Biology: Key Concepts
Explore essential AQA Biology topics including Photosynthesis, Respiration, Homeostasis, Genetics, and Ecology. This comprehensive knowledge organizer covers key concepts such as energy transfer, hormonal control, and genetic variation, providing a solid foundation for your studies. Ideal for exam preparation and understanding biological processes.
109,192314
MathsComprehensive Maths Concepts
Explore essential mathematical concepts including powers, geometry, statistics, and probability. This resource features 65 pages of detailed explanations, diagrams, and examples to enhance your understanding of topics such as right triangles, volume calculations, and data representation. Ideal for students seeking to strengthen their numeracy skills and grasp complex mathematical principles.
1180,3536,327
PhysicsPhysics paper 2 notes
physics aqa gcse paper 2 combined higher notes
111,26910
Biologybiology paper 1
all notes
112,61438
English LanguageLanguage Paper 1 Strategies
Master the AQA English Language Paper 1 with this comprehensive guide. Explore key strategies for language and structural analysis, critical evaluation, and creative writing. Learn how to effectively analyze texts, utilize literary techniques, and enhance your writing skills to excel in your exams.
1128,760902
Chemistrychem paper 1
higher
112,27846
Students love us, and so will you.
4.6/5App Store
4.7/5Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan SiOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha KlichAndroid user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
AnnaiOS user
Maths164 views·Updated 2 Sept 2026·2 pagesThe 7 Circle Theorems Explained with Examples
Circle theorems are essential geometric rules that help you solve angle and length problems involving circles. These theorems appear frequently in GCSE maths exams and once you understand the patterns, they become powerful problem-solving tools.
1
of 2
Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Basic Circle Theorems
Radius and tangent relationships form the foundation of circle geometry. When a radius meets a tangent at the point where it touches the circle, they always form a perfect 90° angle - this is one of the most reliable rules you'll use.
Here's something quite clever about tangents: from any point outside a circle, you can draw exactly two tangent lines. The distances from that external point to where each tangent touches the circle are always equal. This equal tangent lengths rule often helps you find missing measurements.
The angle at the centre theorem shows a beautiful relationship between angles. Any angle formed at the centre of a circle is exactly twice the size of the angle formed at the circumference when both angles use the same arc. This 2:1 ratio appears constantly in circle problems.
Quick tip: Remember "centre doubles circumference" - the angle at the centre is always double the angle at the circumference for the same arc.
Finally, any angle in a semicircle is always 90°. If you spot a triangle where one side is the diameter of a circle, you've found yourself a right-angled triangle.
2
of 2Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Advanced Circle Theorems
Angles in the same segment are always equal to each other. This means if you have multiple angles that sit on the same side of a chord and use the same arc, they'll all have identical measurements - quite handy for solving complex problems.
Cyclic quadrilaterals (four-sided shapes inscribed in circles) follow a special rule: opposite angles always add up to 180°. If you know three angles in a cyclic quadrilateral, you can instantly find the fourth one.
The alternate segment theorem is perhaps the trickiest but most elegant rule. The angle between a tangent and a chord equals the angle in the alternate segment. Think of it as angles "on opposite sides" being equal.
Exam strategy: These theorems often combine in questions, so look for multiple rules working together rather than just one isolated theorem.
Understanding these patterns means you'll recognise the theorems quickly in exam questions. Practice identifying which theorem applies first, then apply the rule confidently.
We thought you’d never ask...
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar content
Most popular content in Maths
9MathsComprehensive Maths Concepts
Explore essential mathematical concepts including powers, geometry, statistics, and probability. This resource features 65 pages of detailed explanations, diagrams, and examples to enhance your understanding of topics such as right triangles, volume calculations, and data representation. Ideal for students seeking to strengthen their numeracy skills and grasp complex mathematical principles.
1180,3536,327
MathsGCSE Maths (Higher) // Revision Guide
The only GCSE maths (higher) revision guide you need to get a grade 9! Contains every topic, each with all potential question types and their solutions.
112,71261
MathsComprehensive Maths Concepts
Explore essential mathematical concepts including polynomial theorems, logarithmic properties, trigonometric functions, and integration techniques. This resource covers everything from solving inequalities to understanding exponential functions, providing a solid foundation for A-level mathematics. Ideal for students aiming for top grades.
1222,0931,822
MathsFoundation Maths Exam Solutions
Explore detailed solutions for the Foundation Tier Non-Calculator Maths exam. This resource covers key concepts such as probability, volume calculations, data representation, and more. Perfect for students preparing for their GCSE Maths exam, with step-by-step explanations and examples.
1021,3821,243
MathsTrigonometric Functions Overview
Explore the fundamentals of trigonometry, including the tangent, sine, and cosine functions. This summary covers key concepts such as SOH CAH TOA, trigonometric ratios, and methods for finding angles and sides in right triangles. Ideal for students preparing for exams or needing a quick reference.
111,77559
MathsGCSE Maths 2018 Exam Insights
Explore the key concepts from the 2018 GCSE Maths Paper 2, including compound interest, probability, standard form, and geometric transformations. This comprehensive summary covers essential topics such as interest rates, area calculations, and Venn diagrams, providing students with a clear understanding of the exam's requirements. Ideal for exam preparation and practice.
98,229411
MathsGCSE Maths Foundation Checklist
Comprehensive revision checklist covering essential topics for the GCSE Maths Foundation tier, including statistics, geometry, algebra, probability, and trigonometry. Perfect for students aiming to pass their exams with confidence.
113,88752
MathsYear 8 Maths AQA Exam
Explore the AQA Year 8 Term 3 Main Paper 2, featuring comprehensive solutions to key mathematical concepts including geometry, percentages, sequences, and data representation. This resource covers essential topics such as area calculations, properties of shapes, and survey analysis, making it ideal for exam preparation and revision.
83,576138
MathsPythagorean Theorem Explained
Explore the Pythagorean Theorem (a² + b² = c²) with clear examples and step-by-step calculations. This summary covers key concepts such as hypotenuse and right triangles, making it essential for understanding geometry. Ideal for students preparing for exams or needing a quick reference.
115775
Most popular content
9SociologySociology of Families: Comprehensive Revision
Dive into an extensive overview of family dynamics, perspectives, and patterns in sociology. This resource covers key concepts such as family diversity, gender roles, marriage, and the impact of social policies on family structures. Perfect for A-Level Sociology students preparing for Paper 2.
1274,0622,307
SociologySociology of Education Overview
Explore comprehensive A-Level Sociology notes on the education system, covering key theories, policies, and sociological perspectives. This resource includes insights on marketisation, gender roles, cultural deprivation, and educational inequalities, providing a thorough understanding of how education shapes social stratification and individual achievement. Ideal for exam preparation and in-depth study.
12103,2233,044
BiologyA-Level Biology Year 1 Overview
Comprehensive summary of AQA A-Level Biology Year 1, covering key topics such as cellular structure, protein synthesis, immune response, gas exchange, and more. Ideal for exam preparation and understanding biological concepts. Includes detailed insights into cellular processes, biological classification, and the circulatory system.
1215,192704
BiologyAQA Biology: Key Concepts
Explore essential AQA Biology topics including Photosynthesis, Respiration, Homeostasis, Genetics, and Ecology. This comprehensive knowledge organizer covers key concepts such as energy transfer, hormonal control, and genetic variation, providing a solid foundation for your studies. Ideal for exam preparation and understanding biological processes.
109,192314
MathsComprehensive Maths Concepts
Explore essential mathematical concepts including powers, geometry, statistics, and probability. This resource features 65 pages of detailed explanations, diagrams, and examples to enhance your understanding of topics such as right triangles, volume calculations, and data representation. Ideal for students seeking to strengthen their numeracy skills and grasp complex mathematical principles.
1180,3536,327
PhysicsPhysics paper 2 notes
physics aqa gcse paper 2 combined higher notes
111,26910
Biologybiology paper 1
all notes
112,61438
English LanguageLanguage Paper 1 Strategies
Master the AQA English Language Paper 1 with this comprehensive guide. Explore key strategies for language and structural analysis, critical evaluation, and creative writing. Learn how to effectively analyze texts, utilize literary techniques, and enhance your writing skills to excel in your exams.
1128,760902
Chemistrychem paper 1
higher
112,27846
Students love us, and so will you.
4.6/5App Store
4.7/5Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan SiOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha KlichAndroid user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
AnnaiOS user