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Maths
14 Dec 2025
421
13 pages
GNisha @gnisha_fhdqlrplxiup
Circles are everywhere in maths, and understanding their equations is crucial for your Higher Maths success. This guide... Show more
Ever wondered how mathematicians describe perfect circles using algebra? The standard form of a circle equation is your starting point , where (a, b) is the centre and r is the radius.
Finding the equation is straightforward when you know the centre and radius. For a circle centred at (2, 1) with radius 7, you'd write . Watch out for negative centres - they become positive in the equation!
You can also determine if a point lies on, inside, or outside a circle by substituting coordinates into the equation. If the result equals , it's on the circle; less than means inside; greater means outside.
Top tip When given a diameter instead of radius, remember to halve it before squaring in your equation!
The general form might look scary, but it's actually quite handy. Your formulae sheet tells you the centre is and radius is .
Checking if it's actually a circle is crucial - you need . If this expression is negative, you don't have a circle at all! This catches out loads of students in exams.
Converting between forms involves identifying the coefficients. In , you get so $g = -4$ and so $f = -5$, giving centre (4, 5).
For point position testing, substitute coordinates into the general equation. Positive result means outside, zero means on the circle, negative means inside.
Remember The general form is particularly useful when dealing with circle intersections and more complex problems!
Finding axis intersections is a common exam question that's easier than it looks. When a circle cuts the x-axis, set and solve the resulting quadratic. For y-axis intersections, set .
Concentric circles share the same centre but have different radii. If one circle has radius 9 and another has half that radius, they're concentric with the smaller having radius 4.5.
Identical circles have the same radius but different centres. When one passes through the origin with centre on an axis, there are usually two possible positions - one positive, one negative.
These context problems often involve distance calculations between intersection points. Once you've found where the circle meets an axis, subtract the coordinates to find lengths.
Exam hack Always sketch the situation when possible - it helps visualise what's happening and prevents silly errors!
Three scenarios can occur when a line meets a circle two intersection points, one point (tangent), or no intersection at all. This mirrors quadratic equations having two roots, one repeated root, or no real roots.
Finding intersection points involves substituting the line equation into the circle equation. For example, if , substitute this into to get , giving .
Tangent identification happens when you get a repeated root . If , there's only one solution, meaning the line touches the circle at exactly one point.
No intersection occurs when the discriminant is negative. Calculate - if it's negative, the line completely misses the circle.
Study tip Practice recognising tangents by looking for perfect square factors or repeated solutions!
Tangent lines touch circles at exactly one point and are perpendicular to the radius at that point. This perpendicular relationship is key to finding tangent equations.
The method involves finding the gradient of the radius from centre to point of contact, then using the fact that perpendicular lines have gradients that multiply to give -1. If the radius has gradient , the tangent has gradient -3.
Special cases occur when the radius is vertical (undefined gradient) - then the tangent is horizontal with equation . Similarly, horizontal radii give vertical tangents.
Proving tangency can be done by substituting a line equation into a circle equation and showing you get a repeated root. This confirms the line touches at exactly one point.
Quick check Always verify your tangent equation by substituting the point of contact - it should satisfy both equations!
Two circles can relate in five different ways, depending on the distance between their centres compared to their radii. Understanding these relationships helps solve complex geometry problems.
External intersection occurs when (circles touch externally) or (circles cross at two points). When , the circles don't meet at all.
Internal relationships happen when one circle sits inside another. If , they touch internally. When , one circle sits completely inside the other without touching.
Distance calculations between centres use the standard distance formula . Compare this with radius sums and differences to determine the relationship.
Visual learning Drawing quick sketches of these relationships helps enormously with understanding and remembering the conditions!
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.
3
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Quotes from every main character
English LiteratureApp Store
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 THE SCHOOLGPT. 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
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 THE SCHOOLGPT. 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
GNisha
@gnisha_fhdqlrplxiup
Circles are everywhere in maths, and understanding their equations is crucial for your Higher Maths success. This guide breaks down everything you need to know about circle equations, from basic centre-radius forms to finding tangents and intersection points.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Ever wondered how mathematicians describe perfect circles using algebra? The standard form of a circle equation is your starting point: , where (a, b) is the centre and r is the radius.
Finding the equation is straightforward when you know the centre and radius. For a circle centred at (2, 1) with radius 7, you'd write . Watch out for negative centres - they become positive in the equation!
You can also determine if a point lies on, inside, or outside a circle by substituting coordinates into the equation. If the result equals , it's on the circle; less than means inside; greater means outside.
Top tip: When given a diameter instead of radius, remember to halve it before squaring in your equation!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The general form might look scary, but it's actually quite handy. Your formulae sheet tells you the centre is and radius is .
Checking if it's actually a circle is crucial - you need . If this expression is negative, you don't have a circle at all! This catches out loads of students in exams.
Converting between forms involves identifying the coefficients. In , you get so $g = -4$ and so $f = -5$, giving centre (4, 5).
For point position testing, substitute coordinates into the general equation. Positive result means outside, zero means on the circle, negative means inside.
Remember: The general form is particularly useful when dealing with circle intersections and more complex problems!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Finding axis intersections is a common exam question that's easier than it looks. When a circle cuts the x-axis, set and solve the resulting quadratic. For y-axis intersections, set .
Concentric circles share the same centre but have different radii. If one circle has radius 9 and another has half that radius, they're concentric with the smaller having radius 4.5.
Identical circles have the same radius but different centres. When one passes through the origin with centre on an axis, there are usually two possible positions - one positive, one negative.
These context problems often involve distance calculations between intersection points. Once you've found where the circle meets an axis, subtract the coordinates to find lengths.
Exam hack: Always sketch the situation when possible - it helps visualise what's happening and prevents silly errors!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Three scenarios can occur when a line meets a circle: two intersection points, one point (tangent), or no intersection at all. This mirrors quadratic equations having two roots, one repeated root, or no real roots.
Finding intersection points involves substituting the line equation into the circle equation. For example, if , substitute this into to get , giving .
Tangent identification happens when you get a repeated root . If , there's only one solution, meaning the line touches the circle at exactly one point.
No intersection occurs when the discriminant is negative. Calculate - if it's negative, the line completely misses the circle.
Study tip: Practice recognising tangents by looking for perfect square factors or repeated solutions!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Tangent lines touch circles at exactly one point and are perpendicular to the radius at that point. This perpendicular relationship is key to finding tangent equations.
The method involves finding the gradient of the radius from centre to point of contact, then using the fact that perpendicular lines have gradients that multiply to give -1. If the radius has gradient , the tangent has gradient -3.
Special cases occur when the radius is vertical (undefined gradient) - then the tangent is horizontal with equation . Similarly, horizontal radii give vertical tangents.
Proving tangency can be done by substituting a line equation into a circle equation and showing you get a repeated root. This confirms the line touches at exactly one point.
Quick check: Always verify your tangent equation by substituting the point of contact - it should satisfy both equations!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Two circles can relate in five different ways, depending on the distance between their centres compared to their radii. Understanding these relationships helps solve complex geometry problems.
External intersection occurs when (circles touch externally) or (circles cross at two points). When , the circles don't meet at all.
Internal relationships happen when one circle sits inside another. If , they touch internally. When , one circle sits completely inside the other without touching.
Distance calculations between centres use the standard distance formula: . Compare this with radius sums and differences to determine the relationship.
Visual learning: Drawing quick sketches of these relationships helps enormously with understanding and remembering the conditions!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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.
3
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Quotes from every main character
English LiteratureApp Store
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 THE SCHOOLGPT. 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
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 THE SCHOOLGPT. 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
Sociology