Circle area by exhaustion (two visual area techniques)
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- Опубліковано 2 жов 2024
- In this short, we show two fascinating methods of determining the area of a circle using the "method of exhaustion." Which one do you like better?
If you like this video, consider subscribing to the channel or consider buying me a coffee: www.buymeacoff.... Thanks!
The top animation unrolls successive circumferences of nested circular shells. In this manner, the circle area gets transformed into the area of a triangle with base 2pi times r and height r.
Here are other versions from me:
• Circle Area by Peeling...
• Circle area from peeli...
To see an alternate (and quite viral) video showing this in stop motion, check out this one from @MinutePhysics : • Proof Without Words: T...
For more information about this construction, see
personal.math....
or check out this nice survey article by David Richeson from the May 2015 issue of The College Math Journal: doi.org/10.416... .
This bottom animation of a classic visual proof showing how to find the area of a circle by using more and more wedges and arranging them in a rectangle.
This proof can be traced to both Satō Moshun and Leonardo da Vinci (see Smith, David Eugene; Mikami, Yoshio (1914), A history of Japanese mathematics, archive.org/de..., page 130-132 and Beckmann, Petr (1976), A History of Pi, St. Martin's Griffin, page 19).
Here are other versions from me:
• Circle Area (classic v...
• Circle Area Derivation...
You can also read more about this in a great NYT article by Steven Strogatz: archive.nytime...
#math #manim #visualproof #proofwithoutwords #circle #circlearea #archimedes #radius #area #areaofcircle #pi #piday #shorts #circle #archimedes #infinite #methodofexhaustion
To learn more about animating with manim, check out:
manim.community
The top one is more unique. Had not seen it displayed that way before, but as a limit, they are both spot-on accurate.
Not very unique, the same demonstration appears in the talmud from the 4th- 5th century (Tractate Succah)
This is just integrating circumference dr
Interesting thing to note, 2 pi r is the formula for circumference
I've always used the rectangle one. Can't help thinking the triangle one might be easier to understand.
@@Kero-zc5tc That's how integrals work
Method of exhaustion sounds like you're tired of explaining the same thing in so many ways
👍😀
Said every algebra teacher ever.
This man didn't just square the circle. He triangled it too!
Hah!
really it was rectangling
But there is no rectangles in this video. Its a trick that takes advantage of the limit in detail to hide imperfections
yes but it's more of a rectangle than it is a square@@cheese0827
"That's it!" >:(
*Triangles and rectangles your circle*
This is probably the answer to why pizzas are shaped like a circle but are placed inside square boxes while they're eaten as triangles.
There are 0 rectangles in this short. Youre being lied to by the creator
@@cheese0827the limit of the bottom does indeed approach a rectangle, it's not the first time I've seen this visualization
@@cheese0827Quite a bold thing to say about a video that is itself a rectangle.
@@cheese0827 ever heard of a limit?
@@cheese0827jesus almighty. you’re skunked, aren’t ya?
It was a day after the AP test in my calc class, so the teacher was just teaching random stuff cause it didn't really matter. At some point, one student said calculus is completely unrelated to all other math, so my teacher just started deriving a bunch of random math formulas we knew using calculus for the next week or so. It was actually pretty interesting.
I found that presenting both together like this, increased their value to three.
👍😀😎
The top method is incorrect, I believe. It is only a coincidence that it gives a correct answer.
@@ChaitanyaTappu
Since this is algebra and this is a proof using unknowns, there are no coincidences. The point of this is that it is a visual proof, obviously a length like circumference can't be stacked to create an area but that doesn't matter to our eyes. Interestingly if you say the length must be infinitely thin then there must also be infinite layers so a triangle will always form.
'Nonsense' is an anagram of 'oneness' - Dancing on your own is probably fun for a few seconds, okay? Boredom is guaranteed. However when we are dancing 'together' the amplification introduces a whole different level of 'energetic' ringing. Wedding bells is one such sound.
@@ChaitanyaTappu Good afternoon. I thought I understood the top method producing the ever decreasing circle = a triangle = .14159. What do you suspect to be the problem as I am genuinely interested to learn?
Newton: hold my calculus
The top one is like a magic 🎉
I dont think anyone ever explained how those formula worked or the reasoning behind them. I really wish they did because the way that i learn is greatly aided by knowing the why.
I just LOVE the top one, feels so surprising yet still intuitive. There is always some clever realization around the corner. Unfortunately, or fortunately, I avoid trying to be clever at my job as a programmer since other people must be able to understand my code, so I can't always practice this type of thinking
The top method is incorrect, I believe. It is only a coincidence that it gives a correct answer.
You almost got me math homework, but not this day!
Top one is a great effective visualization of infinitesimals: each layer is basically a rectangle of length "2πr" and width of "dr". Then you just take the indefinite integral of "2πr dr" to get "πr^2".
I still love that the circumference of a circle radius Pi is 2 Pi squared.
And the area Pi cubed.
👍
@@josh8584 no no it is not gibberish but seems like it. The comment says that if the the radius of the circle is PI then the area will be obviously 2pi*r but since here the radius is PI then the area will be 2*PI². Same thing applies for area: it will be PI³
@@josh8584 I may be extremely pithy, concise, but I don't say gibberish.
@@josh8584 Another person found all the relevant information. The same way an equation is concise.
I will try and expand my responses, and videos, to include more language to guide the reader in a more careful way.
👍🤘🖖
@@josh8584 eliminating extraneous = concise
🖖
Que animación tan perfecta y que bien explicado. Gran trabajo!❤
The bottom one is most intuitive because most can relate to calculating the area of a rectangle.
Brilliant as usual
👍😀
😊 both are amazing
I'm glad that there are different ways to visualize this. I would've never imagined the first one. But then again, I'm not a psychopath. That's some crazy people thing.
Of the two, I thought the triangle visual was more compelling. Though, I do prefer Archimedes approach of inscribing regular n-gons.
The top method is incorrect, I believe. It is only a coincidence that it gives a correct answer.
@@ChaitanyaTappu Well, if that’s the case, then the top method becomes a cautionary tale of why one cannot rely on visuals alone -if even at all.
I like the bottom one better. The dimensions have more clarity.
Its a lie, its no rectangle
@@cheese0827 The limit as you take thinner and thinner slices approaches a rectangle.
Didn't Archimedes invent something like this? So cool!
Showing animation at the bottom while explaining the top was confusing at first.
Lower one is better
It's not a matter of "liking" one more, it's whether the uneducated can grasp the point and see the relation. For those versed in mathematics, the triangle makes sense, as does the rectangle. But for the ignorant, the rectangle is more relatable and comprehensible.
That's enough UA-cam shorts for today
the bottom feels more intuitive for me
your feelings are irrational
@@Fire_Axustheir feelings are irrational, especially since the bottom one is a lie
@@cheese0827 It’s not, though. They’re the exact same.
The triangle circle limit is awesome. Well, they’re both awesome, the circle is just more awesome
The top method is incorrect, I believe. It is only a coincidence that it gives a correct answer.
@@ChaitanyaTappu elaborate
Best one is by calculus 😅
Love you
I'll say both visual proofs rely on a pretty big assumption that the visualisations are indeed correct, and not any other shape. The top one however has an easy resolution, the perimeter scales linearly with the radius, that proves that the triangle is the only possible correct visual.
However, the proof of the bottom one is not so easy. You have to actually prove that the area difference between the wedges and the idealized triangles in the limit goes to zero. I would say the bottom one is not only incomplete but is actively misleading, because the same basic argument can easily create false proofs.
i mean, u could also say the top one relies on assumptions bc u can’t actually “strip away” circumferences one at a time, bc the number of circles within any other circle with the same center is uncountably infinite. but at the end of the day neither of these are really “proofs”, they’re just fun ways to visualize more rigorous limiting processes like integration. so in that sense, starting with a finite partition of the sphere and refining it is more akin to the actual process of integration than the first idea which is more reminiscent of infinitesimals (not rigorous)
The bottom one is very similar to an incorrect visual proof 3B1B did in their "be careful with visual proofs" video
Bottom Is Better For Me
But Both Ways Are Unique And Both Usable
Second is kinda more easy and satisfying
It's Mindblowing! Thanks for your contribution to education!
I was only focused on the illustration above
when the exam questions you about the area of the circle but you dont remember the formula
For me the best way to understand some new concept is to try and create it with as little information as possible about the concept. I didn't knew the volume of sphere so I tried to figure it out on my own by first dividing the sphere into two equal parts and dividing one of those part into infinitely many slices. Since the top most slice would be infinitely small and the second most bottom slice would be infinitely close to the biggest slice(slice whose diameter is equal to sphere's diameter) if I would add those it would become the biggest slice. So I thought that after doing this with all the slices I would have a cylinder whose height is half of radius. so according to me the volume of sphere came out to be πr³ (actual volume = 4/3 πr³). but when I checked, I found out I was wrong. I thought I would figure it out why but I forgot about it untill today when I saw the above visualization. Can you please tell me what was wrong i my method in a video by visualising it properly.
By exhaustion is right... learning the proofs in calculus was far more exciting. (No offense to the video; my calc professor was spectacular)
I like the top one. I never saw it so well explained.
Bottom one explains better why pi is infinite. Upper one makes easier to imagine the circle's area. You see both in same video make you from noob to circle professor.
@@josh8584 🤓
The bottom one feels much more properly defined. What exactly does it mean to peel off the circumferences in the top one? The act of straightening the circumfrences out shouldnt work because of the curvature.
I like the top one best because it's closer to real-life concepts like a rubber band ball or a roll of tape. The bottom one looks like more of an approximation trick you'd find in the maths books, which if you understood those you wouldn't need tricks to begin with :p
the top one is much more satisfying
Why not both? Take the final result of the top method and cut it in half to make two equal triangles. Flip one 180° and align the hypotenuses to form the rectangle of the bottom method.
Follow up on my previous comment. If you take the resulting triangle and cut it in half (vertical cut as the triangle appears) you can then make it into a rectangle and then make it into a square. That's squaring a circle in three steps.
I like the upper, the circle opening like books.
The bottom have to split in half and turn mess my brain, and the triangle is never real triangle.
Pies aren’t square pies are round!
Then you can take the rectangle and turn it into a square and you've squared a circle in two steps.
I think both are so beautyful 😔,
but also in the bottom diagram if you align al wedges to create a line with spikes, the spikes get smaller and smaller until there's no area and just a line with length pi * r^2.
They get thinner not longer
I did this with an onion and it was literally like he said...
a triangle...
Triangle does not seem as obvious as the bottom one
the top one is better
Thank you,sir
Second way
That's smart... 😮
لا يزال التكامل يعمل 😊
The top one feels much more calculus. I'd only seen the bottom one before this.
The top method is incorrect, I believe. It is only a coincidence that it gives a correct answer.
@@ChaitanyaTappu explain.
@@ChaitanyaTappuI disagree. That's a 2d version of the shell method from calculus. It works for any circle.
bro posted two videos in one short to circum-vent it
triangle thing aren't really obvious about being exactly triangle and require extra proof,so second
Take a bow. I will teach this to my daughter
Soo fucking cool
Both have limit at infinity, as expected.
TRIANGLE
I like the top one better. The calculus approach is more clear for this.
The top one is very satisfying.
huh neat 2nd comment
and also 4 minutes from posting
Kinda awesome how an early post got to be recommended by the shorts algorithm, if that happens... you already won the youtube algorithm
2nd one is more easy than 1st😮❤
I too like to unpeel my circles
note that this can go wrong very quickly without being made rigorous by calculus
Wait for tomorrow’s pre pi day short. ;)
brilliant.
Ok, so why is a triangle a linear figure (in area)….. ???…..it is easy to see it is linear in its sides, and so the distances between the sides is also likely to be linear, but more explicit proof is missing. We need ds/dh (s = distance between sides parallel to the base) = constant, h being the height along the radius that becomes the altitude
I’ve seen the one at the bottom before, I like the top one. Thank you, presentation is very clear.
The top method is incorrect, I believe. It is only a coincidence that it gives a correct answer.
too exhausting for me
The lower one is learned and well known, the upper one is completely new for.me and totally excellent. Great video.
Thank you very much
Definitely the triangle
Its just making me realize that all circles have only been approximated in area.
Only an approximation if you don't take the limit at infinity. Using the true limit definition at infinity gives an exact value.
To make it more understandable i'll use algebra
Top:
x=½×2πr×r
x=(½×2)πr×r
x=πr×r
x=πr²
Bottom:
y=πr×r
y=πr²
top one clears
The square one feels slightly more intuitive.
Blue is better.
Exhausting.
The triangle
I prefer the bottom one. It's easier to understand.
3blue1brown made a video on these
For sure. These are classic. But his software library makes it relatively easy to redo :)
The top one
As a kid I thought of the triangle visualization. How did I go from that smart to now needing to ask google how to brush my teeth?
top one is like integrating for area in polar coordinates
Triangle
Both❤️
I just used S=pr, where p is half of perimeter, r is max radius of a circle in a rectangle. In this case rectangle is circle (N→infinity). p=2(pi)*r/2=(pi)*r. S=pi*r², easy
I've seen both, tge 2nd one is def more intuitive
Top
Cool
Top
Top one is more simple.
I guess the top suffers from visualization issue relative to the bottom. If you imagine dividing up the radius by 1/2 you'd get a circle and a donut. Then roll those out straight it'd be a trapezoid of 2 * pi * r base and r / 2 * 2 * pi top. The smaller circle could then be cut in half again giving (2 * pi * r / 2, r / 4 * 2 * pi), etc. Stack all of the those trapezoids together and you get something that resembles a stepped triangle. If you integrate it to infinity by cutting up r /x and you get closer and closer to a triangle of base 2 * pi * r and height of r. The video shows it but doesn't really make it as obvious by doing it step by step (x = 2, x= 3, x=4, etc).
This is Because π = 2 in the Riemann Paradox and Sphere Geometry System Incorporated...
Interior snd exterior polygons area. Exterior square is 4 inscribed is 2. 4 plus 2 fiv by 2 is 3. Pi equals 3.
The assumption that C = kr, where C is the circumference, r is the radius, and k is a constant, does not hold in spherical geometry, for example (such as the geometry where you restrict yourself to stay on the surface of the earth). This assumption is one we make to do Euclidean geometry.
You should've get rid of the phy in the calculations when transforming in triangle and rectangle... but as we cannot do it, it seems that this problem with phy will remain a mistery forever.