Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Geometry
Nov 30, 2025
165
6 pages
aiah @axi4h
Ever wonder how mathematicians describe and measure parts of a circle? Circles are more than just round shapes... Show more
When you draw two radii from the center of a circle, they form a central angle. This angle creates an intercepted arc on the circle. For example, if angle APB measures 95°, the arc AB also measures 95° - the arc measure equals its central angle!
Arcs come in two types minor arcs (less than 180°) and major arcs (greater than 180°). To find the actual length of an arc, you need to calculate what fraction of the circle it represents. The formula is Arc length = × 2πr
💡 Think of arc measure as the "angle size" (in degrees), while arc length is the actual distance along the curved edge (like measuring with a string).
When working with radians instead of degrees, things get simpler! A radian equals the angle that creates an arc equal in length to the radius. For arc length in radians s = rθ, where θ is the angle in radians.
A sector looks like a pizza slice - it's the region bounded by two radii and their intercepted arc. To find its area, use the formula A = × πr², where n is the measure of the central angle.
A segment of a circle is different from a sector. It's the region bounded by an arc and the chord joining the arc's endpoints. To find a segment's area, subtract the area of the triangle from the sector Segment area = Sector area - Triangle area.
Tangent lines touch a circle at exactly one point (the point of tangency). Here's a key rule a tangent line is always perpendicular to the radius at the point of tangency. If you know a line is perpendicular to a radius at the circle's edge, you can be sure it's tangent to the circle.
💡 Tangent segments drawn from the same external point to a circle are always equal in length. This is super helpful for solving many geometry problems!
Chords are segments that connect any two points on a circle. They have several important relationships that help us solve problems
If two chords in a circle are equal in length, then their central angles are equal too. This works both ways - equal central angles mean equal chords! This relationship extends to arcs as well equal arcs have equal chords, and equal chords mean equal arcs.
When working with chords, remember these key relationships to help you solve problems faster. These properties are especially useful when you need to find missing lengths or angle measures in circle problems.
💡 Think of chords like guitar strings - when two strings have the same length on the same guitar, they create similar notes (or in our case, similar measurements in the circle).
The relationship between chords and central angles gives us a powerful tool for analyzing circles, especially when comparing different parts of the same circle or comparing similar circles.
A diameter has special relationships with chords. If a diameter is perpendicular to a chord, it bisects (cuts in half) that chord. This works in reverse too - if a diameter bisects a chord, then it must be perpendicular to that chord.
Another cool fact the perpendicular bisector of any chord always passes through the center of the circle. You can use this to find a circle's center just by constructing perpendicular bisectors of two chords!
Inscribed angles have their vertex on the circle with sides that contain chords. The measure of an inscribed angle is always half the measure of its intercepted arc. This works in all cases, whether the center is on one side of the angle, inside the angle, or outside it.
💡 Here's an amazing fact any angle inscribed in a semicircle is a right angle (90°)! This property is super helpful for finding right angles in geometric problems.
A special property of inscribed quadrilaterals their opposite angles are supplementary, meaning they add up to 180°.
Inscribed angles that intercept the same arc are always equal to each other. This helps us establish relationships between different angles in the same circle.
When a tangent and a chord form an angle, that angle equals half the measure of its intercepted arc. This is similar to the inscribed angle formula but applies specifically to tangent-chord angles.
Secant lines are lines that intersect a circle at two points. When two secants intersect, they form angles with special relationships to the arcs
💡 Notice the pattern? Whether dealing with inscribed angles, tangent-chord angles, or secant intersections, the angle measure is always related to the intercepted arc(s) by some form of division by 2.
These relationships give us powerful tools to find unknown angles when working with circles in geometric problems.
The power of a point refers to an interesting constant relationship when any line is drawn through a fixed point and intersects a circle, the product of the lengths of the two segments from that point to the circle remains constant.
This works in three key situations
For example, if point P is outside the circle and you draw multiple lines through P intersecting the circle, the product of the segments (from P to each intersection point) will be the same for all lines.
💡 The "power of a point" concept is like a hidden superpower in circle geometry! It helps solve complex problems by establishing unexpected relationships between seemingly unrelated segments.
This constant product relationship helps solve many complex geometric problems involving circles and is related to the quadratic formula in algebra.
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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Smart Tools NEW
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
aiah
@axi4h
Ever wonder how mathematicians describe and measure parts of a circle? Circles are more than just round shapes - they have special angles, arcs, and regions that follow precise mathematical rules. Understanding these concepts helps us solve real-world problems from... Show more
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When you draw two radii from the center of a circle, they form a central angle. This angle creates an intercepted arc on the circle. For example, if angle APB measures 95°, the arc AB also measures 95° - the arc measure equals its central angle!
Arcs come in two types: minor arcs (less than 180°) and major arcs (greater than 180°). To find the actual length of an arc, you need to calculate what fraction of the circle it represents. The formula is: Arc length = × 2πr
💡 Think of arc measure as the "angle size" (in degrees), while arc length is the actual distance along the curved edge (like measuring with a string).
When working with radians instead of degrees, things get simpler! A radian equals the angle that creates an arc equal in length to the radius. For arc length in radians: s = rθ, where θ is the angle in radians.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
A sector looks like a pizza slice - it's the region bounded by two radii and their intercepted arc. To find its area, use the formula: A = × πr², where n is the measure of the central angle.
A segment of a circle is different from a sector. It's the region bounded by an arc and the chord joining the arc's endpoints. To find a segment's area, subtract the area of the triangle from the sector: Segment area = Sector area - Triangle area.
Tangent lines touch a circle at exactly one point (the point of tangency). Here's a key rule: a tangent line is always perpendicular to the radius at the point of tangency. If you know a line is perpendicular to a radius at the circle's edge, you can be sure it's tangent to the circle.
💡 Tangent segments drawn from the same external point to a circle are always equal in length. This is super helpful for solving many geometry problems!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Chords are segments that connect any two points on a circle. They have several important relationships that help us solve problems:
If two chords in a circle are equal in length, then their central angles are equal too. This works both ways - equal central angles mean equal chords! This relationship extends to arcs as well: equal arcs have equal chords, and equal chords mean equal arcs.
When working with chords, remember these key relationships to help you solve problems faster. These properties are especially useful when you need to find missing lengths or angle measures in circle problems.
💡 Think of chords like guitar strings - when two strings have the same length on the same guitar, they create similar notes (or in our case, similar measurements in the circle).
The relationship between chords and central angles gives us a powerful tool for analyzing circles, especially when comparing different parts of the same circle or comparing similar circles.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
A diameter has special relationships with chords. If a diameter is perpendicular to a chord, it bisects (cuts in half) that chord. This works in reverse too - if a diameter bisects a chord, then it must be perpendicular to that chord.
Another cool fact: the perpendicular bisector of any chord always passes through the center of the circle. You can use this to find a circle's center just by constructing perpendicular bisectors of two chords!
Inscribed angles have their vertex on the circle with sides that contain chords. The measure of an inscribed angle is always half the measure of its intercepted arc. This works in all cases, whether the center is on one side of the angle, inside the angle, or outside it.
💡 Here's an amazing fact: any angle inscribed in a semicircle is a right angle (90°)! This property is super helpful for finding right angles in geometric problems.
A special property of inscribed quadrilaterals : their opposite angles are supplementary, meaning they add up to 180°.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Inscribed angles that intercept the same arc are always equal to each other. This helps us establish relationships between different angles in the same circle.
When a tangent and a chord form an angle, that angle equals half the measure of its intercepted arc. This is similar to the inscribed angle formula but applies specifically to tangent-chord angles.
Secant lines are lines that intersect a circle at two points. When two secants intersect, they form angles with special relationships to the arcs:
💡 Notice the pattern? Whether dealing with inscribed angles, tangent-chord angles, or secant intersections, the angle measure is always related to the intercepted arc(s) by some form of division by 2.
These relationships give us powerful tools to find unknown angles when working with circles in geometric problems.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The power of a point refers to an interesting constant relationship: when any line is drawn through a fixed point and intersects a circle, the product of the lengths of the two segments from that point to the circle remains constant.
This works in three key situations:
For example, if point P is outside the circle and you draw multiple lines through P intersecting the circle, the product of the segments (from P to each intersection point) will be the same for all lines.
💡 The "power of a point" concept is like a hidden superpower in circle geometry! It helps solve complex problems by establishing unexpected relationships between seemingly unrelated segments.
This constant product relationship helps solve many complex geometric problems involving circles and is related to the quadratic formula in algebra.
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.
0
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
Calculus 1
Algebra 1
Overige