Area of a circle, formula explained
ฝัง
- เผยแพร่เมื่อ 17 มี.ค. 2011
- mathematicsonline.etsy.com
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πr(squared) gives you the area of a circle, but where does it come from? Here is a simple explanation. The area of a circle is the space occupied by the circle in a two-dimensional plane. The space occupied within the boundary/circumference of a circle is called the area of the circle. The formula for the area of a circle is A = πr2, where r is the radius of the circle.
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Quarantine day XX: Learn how to calculate the area of a circle
Quatantine day xii. I am too
same
Lol
its just for my math class that I completely forgot about :(
Ok. For a little excitement differentiate it to get the volume of a sphere. Calculus. Have fun.
This is real mathematics, real understanding.
Angular
React
a colon is :
Mathematics is found in insight, and not in computation. Forgot the mathematician who said that.
Why do schools skip true understanding? math classes today are just; here’s formulas! and rules! now put these 100 problems into your calculators.
For anyone wondering where 2πr came from, π is DEFINED as the ratio of the circumference to the diameter of a circle, π = circumference/diameter
diameter = 2 x radius = 2r
So, circumference/2r = π
Multiply both sides by 2r and you get
Circumference = π x 2r or 2πr
This is not proof or anything, just the fact that π is a special number and a ratio and also an irrational number (cause the ratio of two rationals would never produce an irrational, one of them must be irrational, in this case it's the circumference. Hence it must have π in it's formula)
EXHALER Wolf Still begs the question how to measure the circumference? The whole purpose I thought was not to have pie 🥧 in the final conclusion. 🎰
@@uncleteam A circle is effectively a polygon with infinite sides. First values of π were measured with polygons that had stupid amount of sides. So no matter how many sides you add, you can never truly 'measure' the circumference of a circle. There will always be error. So the best way is to define a ratio of the total perimeter to the longest diameter as some number (π) that approaches a value 3.14159... but never ends or repeats itself.
So the best way to measure circumference of a circle is to measure a quantity that is rational, therefore can be measured with much more greater accuracy, the diameter or radius and use the appropriate formula to get an accurate enough result. Numberphile concluded that only 39 digits after decimal of π is needed to calculate properties of a circle/sphere of size of the known universe with an error comparable to the diameter of a hydrogen atom. Yet our computers have calculated up to trillions of digits after decimal.
It's so intuitive when you use a piece of string to surround the circle and as you spin it into a straight line you just compare to the diameter to realize it's ~=3.14 bigger. The string makes it so practical, you barelly need numbers to work it out, as long as it is proportional.
You still have to demonstrate that the ratio between the circumference of ANY circle and its diameter is a constant (π).
Thanks i need that
I wish that teachers explained it in this way when I was in school. This explanation was very clear... Thanks!
We did this and it was in the book as well
@@mr.schloopka1124 i never learned this until now. Im currently taking calculus 2. more often than not 2 year olds would compare the 4 corners of a square and add up its angles, 360°, to a circle, and jokingly say "sEe? a SquaRe iS eqUAl tO a CiRCle"
@@mr.schloopka1124 yes this was in book but not in this manner
Interestingly, this is basically a calculus approach to the question. Take the integral of the circumference to get the area.
clear explanation + wonderful animation = perfect video!
Quỳnh Huỳnh I agree
Quỳnh Huỳnh agree
+undo.kat you can say, where the base came from, since he compared it with the circle. And you can say, where Pi came from, since everybody learns it in school. You really want the proof of Pi? Ask your teacher about it, since things like Pi, e, Sin, Cos, Tan and co are constants, that no one dares to doubt. lol
Nice explanation
+ Beethoven music in the background 😊
Why the HELL didn't they teach the basis of things like this in High School. It is so much easier to understand the formulas when you can see where they come from.
They did, though. What he is doing is using limits to get a definite integral.
This is all done in highschool, it's just not animated like this.
MSloCvideos wasn’t for me, teachers just tell u it’s pi(r^2) and that’s it, I’ve never seen an explanation before TH-cam
@@MSloCvideos no, teachers tells you the formula, but doesn't explain how you get the formula, which makes it confusing for students.
@@poopswagtyrone7543
This video replicates the original proof given by Archimedes.
@@MSloCvideos - Sounds like you had a teacher that cared to explain why. Many people don't have such thorough teachers.
My whole class found it so satisfying how the triangles just for perfectly together xD
I know, right. Like the proverbial puddle of water fitting into the pothole.
They showed this in your class? Lucky bitch
An elegant, simple and entirely understandable description.
If I'd had you as my teacher, all those years ago, I'd probably have ended up as a mathematician.
And this is what pisses me off about modern education
@@CStrik3r care to explain the other flaws you perceive, as well as your solutions/ideal scenarios?
@@gabrielrabelo6982 teachers vomit formulas and concepts like it is all but a memory game. True learning and understanding comes from logic. Without rational discourse no learning is taking place. But more important than rational teachings is to encourage the student to become an active learner. They must learn how to learn - that is, to seek the truth behind the theoretical framework of readily available conclusions (instead of becoming satisfied with knowing the formula of the area of a circle, to learn the process behind it by their own volition and curiosity, for example). That way, I believe the average student could likely finish learning effectively the entire curriculum behind their college major three to four years earlier than it is expected today (they would already finish highschool content before the age of 15, and finish the equivalent content of a college major by the same time someone today would start college). Sal Khan did exactly that when he was accepted into MIT - in 4 years he completed a triple STEM major in math, CS and electrical engineering by not going to any class at all and just learning by himself through the books. The degree of time effective study an active learner holds just makes for a far better use of their time spent studying than a passive student who keeps expecting to be spoon-fed the knowledge from your usual school and college classes. I myself enrolled in a top business school but I never tried to pick up the reference material up until the last year in college. I started reading from start to finish the main bibliography before the scheduled classes, and kept reading the material instead of attending, and in a week I could cover the whole content from 100 hours of classes and beyond. Many others who lived a similar experience to mine can attest to my following statement - that is, 4 months is more than enough for an active learner to cover the whole finance major, instead of the scheduled time of years.
@@gabrielrabelo6982 ⬆️
@@gaaraio2771 brasil é assim mesmo
This made me understand maths like 10000000x better THANK YOU
niceeeeeeeeeeeeer video
I read: this made me understand maths 100000x better THAN you. lol
@@dekogg lmao, don't think maths as a competition my dude, think it as a topic that we can all share joy with each other
2011 YT: people aren't ready
..
..
..
2020 YT: the time has come!
Because of online class
Your account is only 5 years old. How would you even know if it was recommend in 2011 or not? 🤦🏾♂️😂
@@Until_It_Is_Done Account was made in 2010. You can check view velocity using Vid IQ. or are u indians too poor to afford that shit?
@@bounyh508🤦🏾♂️ click on his name ya idiot! it says "joined 5 years ago". 😂😂 so Tabby boi, are you stupid or just stupid? 🤭
@@Until_It_Is_Done Jesus, are you stupid? It litteraily saids video made in 2011. Are you blind or stupid? Probably both.🙄🤦♂️ Did your parents drop you om the head when you were a kid?
Any legend here watching after 12 years 🤣🔥
Yes I am here 😂😂😂
Me too!
I'm watching in 2025 😂
13 actually
2019: memes
2020: memes and a little bit of math
i was expecting you here, engi
But he used comic sans
Lol
why using comic sans is wrong
comic sans is just tacky, if you still like though I recommend "Comic Neue", it's a more professional looking comic sans
I find this hillarious bcs my math teacher uses comic sans when making our test paper lolol
Fckk
Had I learnt math this way in school..I would have been an physicist.. Absolutely easy to grasp...Fantastic.
3 minutes, easy to follow for anybody, voice is crystal clear, video graphically helps with understanding… 11/10 video.
The beautiful thing about this is, the mathematically rigorous way of solving this question using integration is essentially the same thing. What you did at 1:05 is exactly what a Riemann sum does.
bro our math teacher sent us here
when the teachers get tired of sh!t
Kitcat uwu same
Same
Fucking shit maths
Thank you so much! This was so helpful and I got it right away! Your explanations are very clear
If my teacher teach me in this way, my math will be 100/100
My god pure gold !!! 😮 I was forced to learn the formula , but i coudnt understand where it came from , many thanks for this eye opening class
Las demostraciones geométricas son un lenguaje simple y visual que acompañan a la demostración matemática que es más abstracta y compleja de entender , dándole apoyo a su comprensión y sobre todo a su aplicación práctica en el mundo físico
Thank you so much! The animations helped with the explanation, and the explanations were spoken at a good rate. I admittedly had to rewind the video at about 2:24, but after that, got the concept clearly. Great video!
I discovered this same idea independently when i was at 16 years old in my own interest. I am happy to see this video today. Unfortunately now i am a medical student due to my parents interest!
@Bantai Rapper lol, you just had to one up this guy. Were you jealous of him or something?
Indian, makes sense
@@stuffido1536 pretty sure hes joking lol
The world needs good doctors.
Glad you went the medical rather than academic math route.
I solved it with other way. I found a formula to the area of every regular polygon, adding the areas of the triangles that the polygon is made of. The result: A=d*b*n/2, where (d) is the distance between the centre of the polygon and the centre of one side; (n) is the number of sides; and (b) is the lenght of the side. The area of a circle would be the limit of that formula when (n) tends to infinite, b would be the circunference split for the number of sides, and d would be the radio. If you solve that limit, the answer will be pi*r^2. Sorry if my english isn't good, I'm peruvian. Good video!.
Watching - and working through - your elegant, accessible video was exhilarating. It took me about 15 minutes of noodling around on a piece of paper, exploring your ideas, and I now finally understand in a very pragmatic way the fundamentals of the relationship between pi, circumference, diameter, radius and area. A thousand thanks!
Thank you so much. This helped me understand one of the topics for a test a lot more than I did before. Great video!
This is amazing! I never learnt the origin of area formulas when I was at school but becoming a teacher, they want us to teach the kids this and this is exactly what I was looking for :)
The best visual demonstration of the area rule I have ever seen, fantastic!
Thank you so much for this really easy to follow video. I always struggle understanding concepts because I want to know how we derive the formula. This was very helpful!
This is such an excellent explanation of the Area of a Circle. I show my classes this video each time I teach this topic. Thanks a million!
You're a good teacher if you are looking up how to explain the area of circle to students! I knew the formula but wasn't sure how we arrived at the formula so this video explained it very well. Good luck.
This video literally brought me to tears. Something about deriving the area of circle makes me remember the time I spent with my mother when I was little.
Who got a bigger slice of pizza than?
I'm impressed.
What a simple but thorough explanation.
I love your animations.
Thank you very much for your clear explanation with animation!!! It helped me a lot to fully understand the logic behind the equation. You save me a lot of time figuring it out on my own.
All my years I have taken these wonderful formula for granted. Understanding the hard work done in early times is very interesting and would help understanding calculus down the road.
Nice. Is there a similar way to envision the formula for the circumference, which you use to get the area formula?
First time I’ve ever had it explained as πr times r. Awesome! Thank you for explaining that with visuals - helpful!
This presentation is clear and cool! This is much more important than applying the formula repetitively without knowing the proof in school..
This a strong basic demonstration that appeals to visual intuition. We watched this video in my math methods class, and everyone loved it. Keep up the good work.
Damn! Thank you, I was mistaking the circumference of the circle for the base. Appreciate the effort and clear explanation.
You can do that (base= circumference) but then ur height will be quarter of the diameter (D/4)
Your videos are awesome and very informative and are on a different level from most explanations, Thank You.
I've never had anyone explain it this way, this is too good
dis was really gr88!!! I wish ppl kept making such wondrful vids
+ishan bhange I wish you improved your spelling.
Dis was really greightyeight?
Bruh too many abbreviations. Just use normal English
This video is not explain how come "Circumference =2πr" come from.
It would be complete to prove this sentence.
you must first understand how pi was discovered!!!Go and find on the net how Archimedes managed to calculate pi and this will give you the full answer.
π is just the ratio between the diameter and the circumference. the diameter is 2r. its pretty self explanatory.
π=C/D
C=Dπ
C=2πr
+UbererSK yep
Eliel Dushime
*you must first understand how pi was discovered!!!Go and find on the net how Archimedes managed to calculate pi and this will give you the full answer.*
Archimedes didn't calculate Pi, he just marked the upper and lower limits of its value. Which is something very different from dividing things infinitely and multiply them by infinite.
Jirawat Srisarntiwong by definition, π=circumference/diameter = circumference/2r
ie circumference=2πr
The best explanation for area of circle I have ever seen. The perfect animation is cerry on top.
I learnt this when I was pupil, I still remember the teacher told us use the scissor to cut the circle. A few days later, she brought a cylinder, broke into pieces and combine to a brick. Teaching us how to compute the volume. I have to say the scientific education is so great in China before university
it is easier to understand the area as the addition of all of the circumferances of smaller circle within it: the intergral of 2pie*r with respect to r. but still great video and great channel
No, that is not easier to understand. That requires calculus which many find rather difficult. This video can be shown to a middle schooler and they would understand it. So this video is easier to understand
@@neurofiedyamato8763 that is calculus but not really, what you end up with is a right triangle with base R and height 2(pi)R. You just have to calculate the area based on this.
@@theodiscusgaming3909 how do you prove that the area of the triangle is indeed the area of the circle?
@@MSloCvideos same as the video. The collection of rectangles approaches a triangle as they become smaller and smaller.
@@MSloCvideos You don't. It's an assumption. Or rather, it's a definition. What does the area within a curved region mean? We _define_ the notion of area in a curved region so that this method works. This is an extremely reasonable definition because of what AyyMD VEGA said - we can see that as we increase the number of triangles we use, our intuition of what the error is gets arbitrarily close to 0. We just say, then, that this is what area means.
This made everything so clear for me! It to totally blew my mind XD I mean I had always memorized the rule but I never actually knew why we have that rule. Thank you so much
Dude your the only man that can use a simple voice and music and get me so amazed and invested into math LIKE BE MY TEACHER PLZ
Thank you so much. Great explanation.
I've looked pretty much everywhere to find out the explanation to the formula. I looked on Khan Academy, Google. I found another vid that had the same content of yours but I could not understand it. I come along and see ur vid and the light just penetrates and I see the light to y the formula is. Thank you so much. Please keep making vids like this it really helps out. Do not even look at what people have to say bad about you. I have absolutely no criticism and see no reason y somebody would say anything bad for such a good vid. Pls keep doing more vids like this.
Conclusion: AWESOME VIDEO!!!!
It can also be like - Area of a Circle = Area of a parallelogram = base × height = 2(pi)r/2 × r = (pi)r × r = (pi)r² . It is good for making models because dividing in smaller parts is harder.
❤😂🎉😢😮😅😊
π = 2 in Riemann Paradox And Sphere Geometry Mathematical Systems Incorporated...
This is so clear and helpful. Thank you!
Clear Explaination, Greatly Understood, Perfect Maths, Perfect Learning
awesome video, love it! Thank you for making this video.
Ok now the math is mathing..
I asked this exact question to a friend and struggled with trying to find a answer for quite a lengthy piece of time. Thanks for the explanation now I know :)
I SO LOVE THIS TUTORIAL! Thank you!
p*r^2= 2*integral (sqrt (r^2-x^2))dx from -r to r.
This complets the proof!
@@yesdcotchin you mean dx?
Yep. But this is the same without using integrals.
Can you tell me what software you used to create this video?
I'd love to know this answer too!
Same u would want to know
I*
WOW...THE WAY YOU PRESENTED THE VIDEO IS AMAZING....👍🏻👍🏻:)
Superb visualisation for people who struggle with the 'standard way' of doing math. Well done love it!
What software are you use for making this video?
Please reply
Mind=Blown.
I'm 61 years old and in the length of your video I found myself understanding the subject.
My 12 years at school where wasted on a mind to young to learn and a system not designed to teach true deep understanding.
This is EXACTLY what i've been looking for THX sooooooo much!😁🧡
Bleeping Love it, explained Flawlessly Perfect !
All people here saying pi by definition is the circumference over the diameter but no one is mentioning a proof or reason why that ratio is constant :/
@dalia rosstom I can't just "define"(a very overused abused term) a number to be the ratio between the length to width of a rectangle. There isn't just 1 ratio; a rectangle can have lengths and widths with different ratios relative to one another. Why wouldn't the circle be different? You should prove that the ratio of the circumference to the diameter is always constant.
@@MegaMoh The radius of the circle is always the same
@@MegaMoh pi couldn't change. It will be different only if radius is different. However, that will not longer be a circle, as the definition of circle is "A circle is the set of all points in a plane that are *equidistant* from a given point called the center of the circle"
@@MegaMoh definition of rectangle: "In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal; or a parallelogram containing a right angle."
It only specifies the angle, not any ratio or dimensions. So you can push and pull until length to breadth is 31469420^31469420:1^-1000
but it would still be a rectangle if its angles were the same.
@@deezem5294 you can make a rectangle that has a side's length equal to 5 and then change the other side's length. So it is possible to have a 5x2 rectangle, a 5x3 rectangle, a 5x4 rectangle, a 5xN rectangle where N is just any number. You should prove that a circle with a radius = 5 can not have a circumference of different lengths and can only have 1 length. That is an assumption that you should prove.
Wow! This is an amazing video. This is so clever -- I never thought of it this way!
Thanks a ton! You made my project much more simpler and even easier to understand.
and you also should say what pi actually is which is basically how many times the diameter(radius * 2) fits into the circum. and that ratio is pi and the reason pi is infinite is because "a circle is when a line/multiple lines connected reaches infinity", so the ratio can never be one exact number.
@The Revealer pi actually can not be infinate, only if it's imaginary. Since the angle restriction between 2 each lines will eventually come down to a planck length (1.61622837 × 10^-35 meters).
how complicated it is to find just the area of a circle and need this for grade 7 makes me scared to how complicated math will be in grade 12
So you are in Grade 11 now?
Awesome. These animations in correlation with the narration made this whole thing easy and concise to understand. How were these animations Made? Some animations look CPU/GPU intense.
thank you so much, you taught me more than all my math teachers. honest.
how do u know that the base is half of the circumference if the circle needs to be divided "infinite" times??
Base = infinite?
That "infinite" division means the resulting shape is a rectangle so you can get the area base*height, but the base being half of the circunference comes from the fact that for every slice taken from the circle you take another one in the opposite position
still don't get it, but thx
Because when we divide the circle into triangles we put half of them on top and half on bottom
notice that in 0:12 he puts the lower half of the circle at the bottom of the "rectangle". That lower half is exactly half of the circumference
what's half of infinite?
I learned in a different way
in what way
the kids in my class love this video. You are the greatest youtuber of all time. the end
Wow, thanks!
Thanks a lot i was searching this kind of channel for long
Pie r square. Everyone knows pie is round
🤣
ahhaha
why have I never heard of this. It all makes sense now
Gary Esken
Yes; Pie r round.
Cobbler r rectangle.
now i can die in peace ! :D
but first go ask any or all math & geometry teachers "where does the area of a circle comes from" :D
I had no idea until now. Thank you so much. The light just went on.
Thanks for explanation, this was awesome!
If u had derived circumference ( 2πr ) , then this video would make sense.
shariq ayjaz because 2x radius is diameter and 3.14xdiameter is circumference
The pi is the ratio of circumference of a circle and its diametre. It was practically experimented and this relation was derived by the mathematician in past. Now
Pi = Circumference/diametre
Pi = Circumference/2radius
2pi×radius = circumference or Circumference = 2pi×r
It's not hard to derive it yourself. This video was used to explain the hard part
What this video doesn’t explain, is why the base of the rectangle is equal to half the circumference of the circle. The reason is: the circumference is equal to n times the arc length of each “pie slice” (arc=2πr/n); when these slices are arranged to resemble a rectangle, each long side of the rectangle is made up of n/2 of those arcs; so base=(n/2)*(2πr/n)=πr.
this is Perfect I am goning to Show this video to MY children
Edit: I will show it to my children when I have them (I am 14)
bro u nailed it mine teacher wasnt able to understand me this concept and in just a video of 2 mins u made me clear like ever u are op
Wonderful explanation.Crystall clear.Thank you
Legends come here in 2024
How can someone give a dislike in a video like this? Outstanding explanation! Thanks!
Ame tu explicación, muchas gracias por explicar las cosas tan bien y tan claras
Beautiful explanation and animation. Thank you
Thank a lot. Very good explanation
wow you made that sound so darn easy .. excellent narrative . 76 years old and still learning stuff.
WOOOOOOW, That's so cool, well made and clear!!
Congratulations!!
This is the best explanation I’ve ever gotten
Very beautiful! I was only taught how to derive the area of a circle via calculus. Yes, it made perfect sense, but I've gotta say, this approach is very beautiful ☺️♥️. Nice one, buddy
Great explanation. Thank you!
An elegant explanation. Thank you!
great work, thanks for the knowledge!
the thing i like most about this video is that you don't need the audio to understand it, the visuals are enough
this is very helpful thanks
great visuals and explaining!
this was mind blowing, thankyou so much
Thanks so much man!! You're the greatest!