Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Pre-Cal
Dec 13, 2025
419
7 pages
Notesby_ShamszeyPh @zeyaa
Ever wondered why car wheels, pizzas, and coins all share the same perfect shape? Welcome to the world... Show more
A circle is basically a collection of points that are all the same distance from one center point. Think of it like being on a perfectly round basketball court where everyone stands exactly the same distance from the center.
The center is that fixed point in the middle, while the radius is the constant distance from the center to any point on the circle. Using the distance formula, we can prove that a circle centered at the origin (0,0) has the equation x² + y² = r².
This is called the standard form when the center is at (0,0). It's like the simplest version of a circle equation - clean, straightforward, and easy to work with!
💡 Quick Tip The radius is always positive, so r² will always be a positive number in your equation.
Not all circles sit perfectly at the origin! When a circle's center moves to point (h,k), the equation becomes ² + ² = r². This is still the standard form, just shifted around.
But there's another way to write circle equations called the general form x² + y² + Dx + Ey + F = 0. You get this by expanding the standard form and rearranging everything.
The connection between forms is neat D = -2h, E = -2k, and F = h² + k² - r². These relationships let you convert between standard and general forms easily.
💡 Memory Hack In standard form, if you see ², the center's x-coordinate is +3, not -3!
Here's something cool - not every equation that looks like a circle actually is one! You can check by using this formula D²/4 + E²/4 - F.
If this value is greater than 0, you've got a real circle. If it equals 0, you have a point circle (just a single point). If it's less than 0, the equation has no graph at all - it's mathematically impossible!
Let's see this in action with some examples. For a circle centered at (0,0) with radius 3, you get x² + y² = 9. For center (-2,3) with radius 4, you get ² + ² = 16.
💡 Test Strategy Always check your discriminant when you're unsure if an equation represents a circle!
Converting from standard to general form is like expanding algebra expressions - you just need to be systematic about it. Start with your standard form and use the FOIL method to expand the squared terms.
For example, with center (3,5) and radius 7, you start with ² + ² = 49. Expand each squared term ² becomes x² - 6x + 9, and ² becomes y² - 10y + 25.
After expanding and rearranging, you get x² + y² - 6x - 10y - 15 = 0. The alternative method uses the formulas D = -2h, E = -2k, and F = h² + k² - r² to get the same result faster.
💡 Pro Tip Double-check your arithmetic when expanding - small errors here will mess up your entire equation!
When dealing with negative coordinates for the center, pay extra attention to your signs. For center (-2,-3) with radius 4, the standard form becomes ² + ² = 16.
Remember that becomes , and becomes . When you expand using FOIL, ² gives you x² + 4x + 4, and ² gives you y² + 6y + 9.
The final general form is x² + y² + 4x + 6y - 3 = 0. Using the alternative method D = -2(-2) = 4, E = -2(-3) = 6, and F = (-2)² + (-3)² - 4² = -3.
💡 Sign Alert Negative centers create positive terms in the general form - don't let the double negatives confuse you!
Sometimes you'll need to work backwards from general form to find the center and radius. This process is called completing the square, and it's like solving a puzzle.
Start with x² + y² - 6x - 4y - 3 = 0. Move the constant to the right side, then group x terms and y terms separately. To complete the square for x² - 6x, take half of -6 , square it (getting 9), and add it to both sides.
Do the same for the y terms half of -4 is -2, squared gives 4. You get ² + ² = 16, which tells you the center is (3,2) and radius is 4.
💡 Completing the Square Always add the same number to both sides to keep the equation balanced!
The discriminant test using D²/4 + E²/4 - F is your best friend for determining what an equation represents. Let's test three different scenarios.
For x² + y² + 12y = 0 D=0, E=12, F=0. The test gives 0 + 36 - 0 = 36 > 0, so it's a circle. For x² + y² - 4x + 2y = -5 D=-4, E=2, F=5. The test gives 4 + 1 - 5 = 0, so it's a point circle.
For x² + y² + 4x - 6y + 18 = 0 D=4, E=-6, F=18. The test gives 4 + 9 - 18 = -5 < 0, so it has no graph. This systematic approach prevents you from wasting time trying to graph impossible equations.
💡 Quick Check If your discriminant is negative, don't even try to complete the square - there's no real circle there!
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
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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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Notesby_ShamszeyPh
@zeyaa
Ever wondered why car wheels, pizzas, and coins all share the same perfect shape? Welcome to the world of circles - one of the most fundamental shapes in mathematics! Understanding circles and their equations is crucial for your math journey,... Show more
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A circle is basically a collection of points that are all the same distance from one center point. Think of it like being on a perfectly round basketball court where everyone stands exactly the same distance from the center.
The center is that fixed point in the middle, while the radius is the constant distance from the center to any point on the circle. Using the distance formula, we can prove that a circle centered at the origin (0,0) has the equation x² + y² = r².
This is called the standard form when the center is at (0,0). It's like the simplest version of a circle equation - clean, straightforward, and easy to work with!
💡 Quick Tip: The radius is always positive, so r² will always be a positive number in your equation.
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Not all circles sit perfectly at the origin! When a circle's center moves to point (h,k), the equation becomes ² + ² = r². This is still the standard form, just shifted around.
But there's another way to write circle equations called the general form: x² + y² + Dx + Ey + F = 0. You get this by expanding the standard form and rearranging everything.
The connection between forms is neat: D = -2h, E = -2k, and F = h² + k² - r². These relationships let you convert between standard and general forms easily.
💡 Memory Hack: In standard form, if you see ², the center's x-coordinate is +3, not -3!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Here's something cool - not every equation that looks like a circle actually is one! You can check by using this formula: D²/4 + E²/4 - F.
If this value is greater than 0, you've got a real circle. If it equals 0, you have a point circle (just a single point). If it's less than 0, the equation has no graph at all - it's mathematically impossible!
Let's see this in action with some examples. For a circle centered at (0,0) with radius 3, you get x² + y² = 9. For center (-2,3) with radius 4, you get ² + ² = 16.
💡 Test Strategy: Always check your discriminant when you're unsure if an equation represents a circle!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Converting from standard to general form is like expanding algebra expressions - you just need to be systematic about it. Start with your standard form and use the FOIL method to expand the squared terms.
For example, with center (3,5) and radius 7, you start with ² + ² = 49. Expand each squared term: ² becomes x² - 6x + 9, and ² becomes y² - 10y + 25.
After expanding and rearranging, you get x² + y² - 6x - 10y - 15 = 0. The alternative method uses the formulas D = -2h, E = -2k, and F = h² + k² - r² to get the same result faster.
💡 Pro Tip: Double-check your arithmetic when expanding - small errors here will mess up your entire equation!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When dealing with negative coordinates for the center, pay extra attention to your signs. For center (-2,-3) with radius 4, the standard form becomes ² + ² = 16.
Remember that becomes , and becomes . When you expand using FOIL, ² gives you x² + 4x + 4, and ² gives you y² + 6y + 9.
The final general form is x² + y² + 4x + 6y - 3 = 0. Using the alternative method: D = -2(-2) = 4, E = -2(-3) = 6, and F = (-2)² + (-3)² - 4² = -3.
💡 Sign Alert: Negative centers create positive terms in the general form - don't let the double negatives confuse you!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sometimes you'll need to work backwards from general form to find the center and radius. This process is called completing the square, and it's like solving a puzzle.
Start with x² + y² - 6x - 4y - 3 = 0. Move the constant to the right side, then group x terms and y terms separately. To complete the square for x² - 6x, take half of -6 , square it (getting 9), and add it to both sides.
Do the same for the y terms: half of -4 is -2, squared gives 4. You get ² + ² = 16, which tells you the center is (3,2) and radius is 4.
💡 Completing the Square: Always add the same number to both sides to keep the equation balanced!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The discriminant test using D²/4 + E²/4 - F is your best friend for determining what an equation represents. Let's test three different scenarios.
For x² + y² + 12y = 0: D=0, E=12, F=0. The test gives 0 + 36 - 0 = 36 > 0, so it's a circle. For x² + y² - 4x + 2y = -5: D=-4, E=2, F=5. The test gives 4 + 1 - 5 = 0, so it's a point circle.
For x² + y² + 4x - 6y + 18 = 0: D=4, E=-6, F=18. The test gives 4 + 9 - 18 = -5 < 0, so it has no graph. This systematic approach prevents you from wasting time trying to graph impossible equations.
💡 Quick Check: If your discriminant is negative, don't even try to complete the square - there's no real circle there!
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the 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
App 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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
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