•

Updated Mar 13, 2026

•

5 pages

Circle Theorems Explained: Rules, Examples & Answer PDFs for Class 9 & 10

Shaz

@shaz2007

Circle Theorems: A Comprehensive Guide for Students

This guide provides... Show more

1 / 5

# Circle Theorems

Rule 1:

Rulez:
A

Rule 4:

Tangent meets
Radius at 90°

a

d

B

Tangents
that meet
at a point
are equal in
length
AB =

Additional Circle Theorem: Perpendicular to Chord

This page introduces an additional circle theorem rule that is crucial for solving more complex problems.

Definition: A perpendicular line from the center of a circle to a chord bisects the chord and forms right angles.

This theorem is particularly useful when dealing with problems involving chords and their relationships to the circle's center and circumference.

Highlight: Understanding this theorem can significantly simplify calculations in problems involving chord lengths and angles within circles.

Circle Theorem Questions: Practical Applications

This page presents a series of circle theorem questions and answers, demonstrating how to apply the rules in problem-solving scenarios.

Question involving angles in the same segment:
- Given: Points A, B, C, and D on a circle
- Task: Find angle ACD
- Solution: ACD = 63° (using the "bow tie" theorem)
Question on angles at the center and circumference:
- Given: Points A, B, C, and D on a circle with center O
- Tasks: Find angles x and y
- Solutions: x = 148° (center angle theorem), y = 106° (cyclic quadrilateral theorem)
Complex question combining multiple theorems:
- Given: Circle with center O, tangent ABC, diameter BE
- Tasks: Find angles ABD and DEB
- Solutions: ABD = 54° $tangent-radius theorem$ , DEB = 90° (angle in semicircle theorem)

Example: In question 2, the solution demonstrates how angles at the center are twice the size of angles at the circumference, a key principle in circle theorems explained with examples.

Advanced Circle Theorem Problems

This page presents more challenging circle theorem questions, requiring the application of multiple rules and deeper analytical thinking.

Problem involving isosceles triangles and tangents:
- Given: Circle with center O, points A, B, C, and tangent XCY
- Task: Find angle OCB
- Solution: OCB = 27° (using alternate segment theorem and isosceles triangle properties)
Complex tangent problem:
- Given: Circle with center O, tangents PA and PB, angle APB = 86°
- Task: Find angle x
- Solution: x = 43° (applying multiple theorems including tangent properties and isosceles triangles)

Highlight: These problems demonstrate how complex circle theorem problems and solutions often require a step-by-step approach, combining multiple theorems to reach the final answer.

Proof and Advanced Applications of Circle Theorems

This page focuses on proving circle theorems and solving highly complex problems, ideal for advanced students and GCSE circle theorems preparation.

Proof question:
- Task: Prove that angle ROS = 2x, given RST = x
- Solution: Uses tangent-radius theorem, isosceles triangle properties, and angle sum theorems
Advanced application question:
- Given: Circle with center O, tangent PT, straight line SOP, angle OPT = 32°
- Task: Find angle x
- Solution: x = 29° (applying multiple theorems and angle calculations)

Example: The proof question demonstrates how to construct a formal geometric proof using circle theorem rules, an essential skill for advanced mathematics.

Highlight: These problems are excellent practice for students preparing for difficult circle theorem questions and answers in exams.

Circle Theorems: Fundamental Rules and Applications

This page introduces eight essential circle theorem rules, providing a foundation for understanding geometric relationships within circles.

Definition: Circle theorems are geometric principles that describe relationships between angles, lines, and points in and around circles.

Tangents meeting at a point are equal in length.
A tangent meets a radius at a 90° angle.
Two radii form an isosceles triangle.
Opposite angles in a cyclic quadrilateral add up to 180°.
The angle at the center is twice the size of the angle at the circumference.
Angles in the same segment are equal.
The angle at the circumference in a semicircle is 90°.
Alternate segment theorem: Angles in alternate segments are equal.

Highlight: These rules are often combined in complex geometric problems, requiring students to apply multiple theorems simultaneously.

Example: Rule 2 and Rule 3 are frequently seen together in diagrams, where a tangent forms a right angle with a radius, and two radii create an isosceles triangle within the circle.

We thought you’d never ask...

What is the Knowunity AI companion?

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.

Where can I download the Knowunity app?

You can download the app from Google Play Store and Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Most popular content

An Inspector Calls: Character Insights

Explore in-depth analysis and key quotes for characters in J.B. Priestley's 'An Inspector Calls'. This resource covers Gerald Croft, Inspector Goole, Sheila Birling, Mrs. Birling, Eric Birling, and Eva Smith, focusing on themes of class, gender roles, and social responsibility. Ideal for students aiming for Grade 8 and above.

English Literature

Sociology 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.

Sociology

Comprehensive Crime & Deviance Overview

Explore an extensive revision of crime and deviance topics, including theories, types of crime, and the impact of media. This resource covers key concepts such as Marxism, functionalism, gender and crime, and the influence of globalization on criminal behavior. Ideal for students seeking a thorough understanding of criminology and its various theories. Type: Full Topic Revision.

Sociology

Sociology 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.

Sociology

WJEC Unit 4 Criminology

Criminology unit 4 detailed revision note

Criminology

aqa higher tier combined science - biology b1 : cell biology, paper 1

b1: cell biology - paper 1

Biology

Criminology: Crime & Punishment Overview

Comprehensive mindmaps covering key concepts in the Crime and Punishment topic for WJEC Criminology Unit 4. This resource includes detailed insights into the Criminal Justice System, crime prevention strategies, sentencing models, and the roles of various agencies. Ideal for A-Level revision, ensuring you grasp essential theories and legislative processes to excel in your exams.

Criminology

Macbeth: Guilt and Ambition

Explore the complex themes of guilt and ambition in Shakespeare's 'Macbeth'. This analysis covers key characters, including Macbeth and Lady Macbeth, their moral dilemmas, and the tragic consequences of their ambition. Ideal for students studying character motivations, thematic elements, and the psychological impact of power. Includes insights on the natural order, manipulation, and the descent into madness.

English Literature

AQA 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.

Biology

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.6/5

App Store

4.7/5

Google 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 S

iOS 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 Klich

Android 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.

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Basil

Android user

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Rohan U

Android user

Xander S

iOS user

Elisha

iOS user

Paul T

iOS user

Maths

•

534

•

Updated Mar 13, 2026

•

5 pages

Circle Theorems Explained: Rules, Examples & Answer PDFs for Class 9 & 10

Shaz

@shaz2007

Circle Theorems: A Comprehensive Guide for Students

This guide provides a detailed explanation of circle theorems, essential for geometry studies in mathematics. It covers eight fundamental rules with examples and practice questions.

Covers key circle theorems explained with examples
Includes ... Show more

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Additional Circle Theorem: Perpendicular to Chord

This page introduces an additional circle theorem rule that is crucial for solving more complex problems.

Definition: A perpendicular line from the center of a circle to a chord bisects the chord and forms right angles.

This theorem is particularly useful when dealing with problems involving chords and their relationships to the circle's center and circumference.

Highlight: Understanding this theorem can significantly simplify calculations in problems involving chord lengths and angles within circles.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Circle Theorem Questions: Practical Applications

This page presents a series of circle theorem questions and answers, demonstrating how to apply the rules in problem-solving scenarios.

Question involving angles in the same segment:
- Given: Points A, B, C, and D on a circle
- Task: Find angle ACD
- Solution: ACD = 63° (using the "bow tie" theorem)
Question on angles at the center and circumference:
- Given: Points A, B, C, and D on a circle with center O
- Tasks: Find angles x and y
- Solutions: x = 148° (center angle theorem), y = 106° (cyclic quadrilateral theorem)
Complex question combining multiple theorems:
- Given: Circle with center O, tangent ABC, diameter BE
- Tasks: Find angles ABD and DEB
- Solutions: ABD = 54° $tangent-radius theorem$ , DEB = 90° (angle in semicircle theorem)

Example: In question 2, the solution demonstrates how angles at the center are twice the size of angles at the circumference, a key principle in circle theorems explained with examples.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Advanced Circle Theorem Problems

This page presents more challenging circle theorem questions, requiring the application of multiple rules and deeper analytical thinking.

Problem involving isosceles triangles and tangents:
- Given: Circle with center O, points A, B, C, and tangent XCY
- Task: Find angle OCB
- Solution: OCB = 27° (using alternate segment theorem and isosceles triangle properties)
Complex tangent problem:
- Given: Circle with center O, tangents PA and PB, angle APB = 86°
- Task: Find angle x
- Solution: x = 43° (applying multiple theorems including tangent properties and isosceles triangles)

Highlight: These problems demonstrate how complex circle theorem problems and solutions often require a step-by-step approach, combining multiple theorems to reach the final answer.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Proof and Advanced Applications of Circle Theorems

This page focuses on proving circle theorems and solving highly complex problems, ideal for advanced students and GCSE circle theorems preparation.

Proof question:
- Task: Prove that angle ROS = 2x, given RST = x
- Solution: Uses tangent-radius theorem, isosceles triangle properties, and angle sum theorems
Advanced application question:
- Given: Circle with center O, tangent PT, straight line SOP, angle OPT = 32°
- Task: Find angle x
- Solution: x = 29° (applying multiple theorems and angle calculations)

Example: The proof question demonstrates how to construct a formal geometric proof using circle theorem rules, an essential skill for advanced mathematics.

Highlight: These problems are excellent practice for students preparing for difficult circle theorem questions and answers in exams.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Circle Theorems: Fundamental Rules and Applications

This page introduces eight essential circle theorem rules, providing a foundation for understanding geometric relationships within circles.

Definition: Circle theorems are geometric principles that describe relationships between angles, lines, and points in and around circles.

Tangents meeting at a point are equal in length.
A tangent meets a radius at a 90° angle.
Two radii form an isosceles triangle.
Opposite angles in a cyclic quadrilateral add up to 180°.
The angle at the center is twice the size of the angle at the circumference.
Angles in the same segment are equal.
The angle at the circumference in a semicircle is 90°.
Alternate segment theorem: Angles in alternate segments are equal.

Highlight: These rules are often combined in complex geometric problems, requiring students to apply multiple theorems simultaneously.

Example: Rule 2 and Rule 3 are frequently seen together in diagrams, where a tangent forms a right angle with a radius, and two radii create an isosceles triangle within the circle.