Circles are everywhere around us - from wheels to plates... Show more
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G
grd 11&12 notes
12/20/2025
Pre-Cal
PRE-CALCULUS
1,118
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Dec 20, 2025
•
Understanding Pre-Calculus Concepts: Circles and Parabolas
G
grd 11&12 notes
@vanethmina
1 / 10
Getting Started with Circles
Your teacher Sir Lloydiee wants you to succeed in understanding circles in precalculus! This topic might seem challenging at first, but with practice, you'll find it's actually quite logical.
Think of circles as the foundation for many real-world applications. Once you understand how to work with circle equations, you'll have powerful tools for solving geometry problems.
Remember: Every circle problem follows the same basic principles - find the center and radius, then apply the right formula!
What Exactly is a Circle?
A circle is basically the set of all points that are exactly the same distance from one central point. That central point is called the center, and the distance from center to edge is the radius.
You can also think of circles as conic sections - they're formed when you slice through a cone with a plane that's perfectly perpendicular to the cone's axis. Pretty cool, right?
The key thing to remember: every point on a circle is exactly the same distance from the center. This simple fact is what makes all circle math work!
Fun Fact: The radius is like the circle's "signature" - it determines everything about the circle's size!
Standard Form of Circle Equations
The standard form is your best friend when working with circles: r² = ² + ². Here, (h, k) represents the center coordinates and r is the radius.
When the circle is centered at the origin (0, 0), the equation becomes super simple: r² = x² + y². This happens when both h and k equal zero.
This formula comes directly from the distance formula - it's measuring the distance from any point (x, y) to the center (h, k). Once you see this connection, circle problems become much easier!
Pro Tip: Always identify the center and radius first - they're the keys to solving any circle problem!
Working with Standard Form Examples
Let's practice! For a circle with center (0, 0) and radius 5, you get: x² + y² = 25. Notice how r² becomes 25 since 5² = 25.
For center (-4, 5) and radius 4: ² + ² = 16. See how the signs change? When h is negative, you add in the equation.
When you're given diameter endpoints like A(4,5) and B(-2,3), first find the center by averaging the coordinates, then calculate the radius using the distance formula. The center would be (1, 4) and radius would be √13.
Key Strategy: When converting from center-radius info to equations, be extra careful with positive and negative signs!
Reading Circle Equations
Going backwards is just as important! From ² + ² = 25, you can read: center at (-9, 1) and radius 5.
For x² + ² = 16/9, the center is (0, 1/4) and radius is 4/3. Remember to take the square root of the right side to find the radius!
With ² + y² = 15, you get center (6, 0) and radius √15. When a variable doesn't have anything added or subtracted, that coordinate of the center is zero.
Quick Check: Always verify your radius is positive and makes sense for the problem context!
General Form of Circle Equations
Sometimes circles appear in general form: x² + y² + Dx + Ey + F = 0. This looks messier than standard form, but it contains the same information.
You get general form by expanding the standard form equation. Take ² + ² = r² and multiply everything out, then move all terms to one side.
For example, a circle with center (-1, -6) and radius 5 becomes: x² + y² + 2x + 12y + 12 = 0 in general form. The process involves expanding, combining like terms, and rearranging.
Remember: General form always has x² and y² with coefficient 1, plus linear terms and a constant!
Converting Between Forms
To go from general form back to standard form, you'll use completing the square. This is like reverse engineering the equation to find the center and radius.
The process has four main steps: divide by the coefficient of x² (if it's not 1), group x and y terms, complete the square for both variables, then factor and simplify.
For x² + y² + 10x - 6y - 2 = 0, you'd group terms, add appropriate values to complete squares, and end up with center (-5, 3) and radius 6.
Master This: Completing the square is your superpower for converting messy circle equations into useful standard form!
Practice Makes Perfect
When you see 2x² + 2y² - 8x - 4y - 8 = 0, don't panic! First divide everything by 2 to get standard coefficients for x² and y².
After dividing: x² + y² - 4x - 2y - 4 = 0. Now group your x terms and y terms, complete the square for each, and you'll find the center and radius.
The more you practice these conversions, the faster you'll recognize patterns. Soon you'll be solving circle problems like a pro!
Success Tip: Work through problems step-by-step and double-check your algebra - small mistakes can throw off your final answer!
We thought you’d never ask...
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Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
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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
Understanding Pre-Calculus Concepts: Circles and Parabolas
G
grd 11&12 notes
@vanethmina
Circles are everywhere around us - from wheels to plates to the moon! In math, circles are special shapes with unique properties that follow specific equations and formulas you can master.
Getting Started with Circles
Your teacher Sir Lloydiee wants you to succeed in understanding circles in precalculus! This topic might seem challenging at first, but with practice, you'll find it's actually quite logical.
Think of circles as the foundation for many real-world applications. Once you understand how to work with circle equations, you'll have powerful tools for solving geometry problems.
Remember: Every circle problem follows the same basic principles - find the center and radius, then apply the right formula!
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What Exactly is a Circle?
A circle is basically the set of all points that are exactly the same distance from one central point. That central point is called the center, and the distance from center to edge is the radius.
You can also think of circles as conic sections - they're formed when you slice through a cone with a plane that's perfectly perpendicular to the cone's axis. Pretty cool, right?
The key thing to remember: every point on a circle is exactly the same distance from the center. This simple fact is what makes all circle math work!
Fun Fact: The radius is like the circle's "signature" - it determines everything about the circle's size!
Sign up to see the contentIt's free!
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By signing up you accept Terms of Service and Privacy Policy
Standard Form of Circle Equations
The standard form is your best friend when working with circles: r² = ² + ². Here, (h, k) represents the center coordinates and r is the radius.
When the circle is centered at the origin (0, 0), the equation becomes super simple: r² = x² + y². This happens when both h and k equal zero.
This formula comes directly from the distance formula - it's measuring the distance from any point (x, y) to the center (h, k). Once you see this connection, circle problems become much easier!
Pro Tip: Always identify the center and radius first - they're the keys to solving any circle problem!
Sign up to see the contentIt's free!
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Working with Standard Form Examples
Let's practice! For a circle with center (0, 0) and radius 5, you get: x² + y² = 25. Notice how r² becomes 25 since 5² = 25.
For center (-4, 5) and radius 4: ² + ² = 16. See how the signs change? When h is negative, you add in the equation.
When you're given diameter endpoints like A(4,5) and B(-2,3), first find the center by averaging the coordinates, then calculate the radius using the distance formula. The center would be (1, 4) and radius would be √13.
Key Strategy: When converting from center-radius info to equations, be extra careful with positive and negative signs!
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Reading Circle Equations
Going backwards is just as important! From ² + ² = 25, you can read: center at (-9, 1) and radius 5.
For x² + ² = 16/9, the center is (0, 1/4) and radius is 4/3. Remember to take the square root of the right side to find the radius!
With ² + y² = 15, you get center (6, 0) and radius √15. When a variable doesn't have anything added or subtracted, that coordinate of the center is zero.
Quick Check: Always verify your radius is positive and makes sense for the problem context!
Sign up to see the contentIt's free!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
General Form of Circle Equations
Sometimes circles appear in general form: x² + y² + Dx + Ey + F = 0. This looks messier than standard form, but it contains the same information.
You get general form by expanding the standard form equation. Take ² + ² = r² and multiply everything out, then move all terms to one side.
For example, a circle with center (-1, -6) and radius 5 becomes: x² + y² + 2x + 12y + 12 = 0 in general form. The process involves expanding, combining like terms, and rearranging.
Remember: General form always has x² and y² with coefficient 1, plus linear terms and a constant!
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By signing up you accept Terms of Service and Privacy Policy
Converting Between Forms
To go from general form back to standard form, you'll use completing the square. This is like reverse engineering the equation to find the center and radius.
The process has four main steps: divide by the coefficient of x² (if it's not 1), group x and y terms, complete the square for both variables, then factor and simplify.
For x² + y² + 10x - 6y - 2 = 0, you'd group terms, add appropriate values to complete squares, and end up with center (-5, 3) and radius 6.
Master This: Completing the square is your superpower for converting messy circle equations into useful standard form!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Practice Makes Perfect
When you see 2x² + 2y² - 8x - 4y - 8 = 0, don't panic! First divide everything by 2 to get standard coefficients for x² and y².
After dividing: x² + y² - 4x - 2y - 4 = 0. Now group your x terms and y terms, complete the square for each, and you'll find the center and radius.
The more you practice these conversions, the faster you'll recognize patterns. Soon you'll be solving circle problems like a pro!
Success Tip: Work through problems step-by-step and double-check your algebra - small mistakes can throw off your final answer!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
We thought you’d never ask...
What is the Knowunity AI companion?
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.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the 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.
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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