Simple Pythagorean Dissection Proof

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Simple Pythagorean Dissection Proof

Рет қаралды 341,102

Күн бұрын

This is a short, animated (wordless) visual proof of the Pythagorean theorem (the right triangle theorem) using a relatively elementary dissection proof.
If you like this video, consider subscribing to the channel or consider buying me a coffee: www.buymeacoff.... Thanks!
This animation is based on a proof from Grégoire Nicollier, Aljoša Peperko & Janez Šter in the December 2018 issue of Mathematics Magazine (doi.org/10.108... ) pages 382-383.
For other proofs of this same fact check out:
• Pythagorean Theorem I ...
• Pythagorean Theorem II...
• Pythagorean Theorem II...
• Pythagorean Theorem IV...
• Pythagorean Theorem V ...
• Pythagorean Theorem VI...
• Pythagorean Theorem VI...
• Pythagorean Theorem VI...
#math #manim #pythagoreantheorem #pythagorean #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof #mathshorts #mathvideo
To learn more about animating with manim, check out:
manim.community

Пікірлер: 262

@Sushrut1101 Жыл бұрын

This is the best proof of the Pythagoras theorem I've ever seen!!

@nuclearwarhead9845 9 ай бұрын

Another cool one is making 4 a-b-c triangles so that they touch on the edges and are oriented perpendicular, this creates a big square with side a+b and a small square with side c, so (a+b)^2-triangles=c^2, and the triangles each have an area of ab/2 so a^2+b^2+2ab-2ab=c^2

@nealjroberts4050 9 ай бұрын

@@nuclearwarhead9845 That's the simplest one I think that demonstrates the theorem as it easily shows the relationship without doing the cuts in the video.

@Mubashar783 9 ай бұрын

Which Software is this? please tell me name 😢

@gustavo9758 8 ай бұрын

Amazing, my 6th grade ass would have been blown away even more.

@Trombonauta 6 ай бұрын

Mathologger has 2 great videos about the theorem too, referencing also a book with hundreds of different proofs for it.

@MemekingJag 6 ай бұрын

i love these visual proofs, i am awful at math but these are so intuitive

@ltpetrenko 5 ай бұрын

You might want to revisit those lies told to you. It took me a few tries to get it, and my PhD didn’t help me much. Lol. Most of math is common sense. It just takes patience and a good tutor sometimes. Should be easy for you, if you got this proof.

@156_____11 Ай бұрын

Most is patience and understanding the concept after a bit. You will get faster. I went from my mom screaming at me after four hours of trying to teach me clocks and basic money and change at age six to a high schooler with a lot of intuition for theoretical math and stats. This came after experimenting with stuff like algorithmic programming (which I realize was NOT for me) and then machine learning (which was). Keep practicing and pursue what interests you!

@ayuballena8217 Ай бұрын

@@ltpetrenkoi argee

@spyresoblazx Жыл бұрын

That's a clever one! Using the triangle as extra space

@YK_data 11 ай бұрын

The first unshaded area is equal to the sum of the second unshaded areas, because, the overall shape is not changed and the shaded areas only changed position within the overall area. This could have mentioned to be clearer. I like the other proof in which you can copy three more of the triangle and rotate each 90 180 270 degrees then bring them together to form a square with side length c.

@wyattstevens8574 5 ай бұрын

Me too!

@KeithJosephs 5 ай бұрын

Agree

@theteleportercell6749 Жыл бұрын

Love these videos, ur so underrated!

@Dante-420 5 ай бұрын

The best visual proof of the pythagorean theorem I've ever seen. You can clearly see how to reconstitute the new shaded areas. Much better than those demonstrations with water that don't really prove the theorem except for that specific triangle

@ramakrishnagodugunur195 8 ай бұрын

I keep thinking about this beautiful explanation often. My love for math has grown because of explanations like these. Great Channel! Thanks a lot!

@MathVisualProofs 8 ай бұрын

Thanks for visiting!

@ThePanman48 28 күн бұрын

There is a light that just came on in my head. 🤯 Visualizing the squared and not just memorizing a formula.

@BlooGekko 9 ай бұрын

I've never thought of it this way before. Awesome.

@user-vk3li2lg7d 11 ай бұрын

Thanks bro I never thought in that way

@RandyKing314 Жыл бұрын

nice! i honestly didn’t visualize that result until the end!

@PHyN1151 9 ай бұрын

But it doesnt mean c²=a²+b² Its means, c²+abc(the first triangle) = a²+b² Correct Me if im wrong Edit: Ooo, it Also use that abc triangle! So its, c²+abc = a²+b²+abc So we can remove the abc triangle from both side, c² = a²+b²

@zhigangxu2007 9 ай бұрын

I was the puzzled too! The author seems did not finish his explanation nicely, leaving the last bit confusing.

@Mubashar783 9 ай бұрын

@@zhigangxu2007in which software it was created please tell me, I am curious to create like this one?

@ayaanzaveri9969 8 ай бұрын

@@Mubashar783It’s a programming language called Manim created by 3Blue1Brown

@Mubashar783 8 ай бұрын

@@ayaanzaveri9969 JazakAllah Brother 😇

@Tusharplays69 8 ай бұрын

And also that's not abc because the area enclosed by Δabc is (1/2)ab

@tuccig0813 9 ай бұрын

Honesty a sick proof that I've never seen before. Idk why the bgm is holding a funeral tho

@nilsalmgren4492 2 ай бұрын

My favorite proof is the a+b by a+b square. Connect the points where a and b meet with the same point on adjacent sides. Call the length of the hypotenuse created c. This forms four axb right triangles and a center square side c. Calculate the area as (a+b)^2 = 4(1/2)ab + c^2 and do the algebra you result in a^2 + b^2 = c^2

@hubertkaczmarek2323 Жыл бұрын

Wonderful, thank You 🎉

@MathVisualProofs Жыл бұрын

Thank you too!

@leif1075 Жыл бұрын

@@MathVisualProofswho came up with this proof? You??

@MathVisualProofs Жыл бұрын

@@leif1075 not me. Check the description. It has the original published version.

@mohammedal-haddad2652 Жыл бұрын

I was curiously looking downstairs waiting what will happen, but suddenly the results appeared upstairs!

@pietro5266 Ай бұрын

Excellent proof; I've never seen this one.

@vipinkumar5955 Жыл бұрын

Lol i thought atleast i knew Pythagoras theorem 😂 u destroyed it with making it more difficult for me

@Steshan 10 ай бұрын

it’s super easy honestly

@AstroEli133 9 ай бұрын

How did a proof change your understanding of a simple equation?

@TheNinthGenerarion 8 ай бұрын

Basically, the area coloured in = triangle + c^2 = a^2 + b^2 + triangle, because originally we had the triangle and the 2 squares, then fit them into the space for the triangle and c^2. We can remove the triangle from both sides to leave just c^2 = a^2 + b^2

@josephcoon5809 8 ай бұрын

Draw in the lines of the original unshaded area. You have a part of a^2 and part of b^2 outside of that original unshaded area.

@MathVisualProofs 8 ай бұрын

This is about the unshaded area in the diagram. It starts as c^2 and ends as a^2+b^2… the covering up of c square is irrelevant.

@josephcoon5809 8 ай бұрын

@@MathVisualProofs The covering up of what a^2 + b^2 equals is irrelevant? Interesting.

@MathVisualProofs 8 ай бұрын

@@josephcoon5809 well what’s irrelevant is *how* it is covered. It seems you are mistaking unshaded are for shaded area. If you prefer shaded area then the diagram starts with a^2+b^2 +1/2(ab) shaded and ends with c^2+1/2(ab) shaded. But it’s cleaner to look at unshaded area in my opinion

@josephcoon5809 8 ай бұрын

@@MathVisualProofs The problem is that you covered it with shapes that are undefined while hoping people can easily remember what each shape represents. The only thing that was easily remembered was the three squares a^2, b^2, and c^2.

@kirkdoray3393 7 ай бұрын

It took me a moment to understand that the 'shaded area' includes the original triangle, in the beginning and the end.

@stardufs 4 ай бұрын

this is mind-blowing

@MathVisualProofs 4 ай бұрын

👍

@bigedwerd Жыл бұрын

There's a cool visual experiment you can do with water in the a and b squares that fills into the c square. I once made a math app trying to convey this but simulating water is hard 😅

@bijipeter1471 6 ай бұрын

Thank you,sir

@MathVisualProofs 6 ай бұрын

Most welcome

@user-sv2ej4hw6c 5 ай бұрын

Because we rotated the original triangle into the C Square, what we're left with is. C squared plus the Area of the original Triangle. Makes it more confusing because the original triangle is not part of the solution.

@thatphatbaby 5 ай бұрын

This is the only thing lacking from the explanation took me a minute to realize that you can subtract that area back out to be left with the pythagorean thrm

@user-sv2ej4hw6c 5 ай бұрын

any baby that be phat will tell yell ya, an elegant visual representation would leave our home triangle unshaded. for real

@erinmac4750 11 ай бұрын

This is not only an elegant proof of the Pythagorean Theorem, but a beautiful example of rotation and translation.📐😎

@jackhenderson9872 5 ай бұрын

That's Beautiful.

@SalahEddineSTM 5 ай бұрын

Thank you so much

@user-jr1ke5im2s 7 ай бұрын

Great job

@Andrej..... 8 ай бұрын

Great visualisation! Love it!

@vidterparia8560 2 ай бұрын

Brilliant!!!

@Allen2 4 ай бұрын

And Pythagoras did all this and more without sharp scissors ... a genius.

@pearhams2 9 ай бұрын

The red square covers the the initial purple triangle's space plus the extra needed to fill in the rest of the C square after the other parts are translated in fully. So the purple triangle chops up the a square and b square into translatable portions that fit into the c square. They could have just as easily slid the triangle into the a square as well and use the leftover to fill the c square. It cuts the a square exclusive and the b square inclusive to fit inside the c square.

@Mubashar783 9 ай бұрын

HY Dear pleases tell me, in which software it was created please tell me, I am curious to create like this one? pls sir😢

@uncouthyouth2433 9 ай бұрын

Brilliant intuitive proof Once it clicks, its awesome.

@sh1nryel754 6 ай бұрын

The problem is you can't really understand or prove that this works unless you actually use physical cards/shapes in an outline or just use a simulation. There are other ways to picture it where you can tell intuitively that they should be the same size.

@raymitchell9736 2 ай бұрын

It's a bit of a messy visual proof, but interesting how it works!

@LexiLex421 4 ай бұрын

How did I sort of know because of those mini I ready animations (not trying to shame you. Great work!)

@Alex_Cara77 9 ай бұрын

You cannot even see a squate with side c, there are many visualization that you can and they make much more sense

@boutiquemalu6500 9 ай бұрын

I love the Pythagorean theorem and i feel proud to actually be a greek who is from pythagorio Greece the city named after Pythagoras who lived there

@zaaaaaaac98 4 ай бұрын

Why is KZfaq literally better than university😂

@kartik-mw2vs 11 ай бұрын

UNBELIEVABLE!!!!❤

@smalin 11 ай бұрын

I see how it works for this particular right triangle, but how does this prove that it will work for every right triangle?

@ceoofbeingdumb 9 ай бұрын

I can't prove it but I will tell you this. Each time you add more to a, c becomes longer. Each time you add more to b, c will also become longer. So anytime a/b is longer, c will also be longer. Thus, the extra area in a²/b² will still fit in c² because c² itself, will increase area perfect enough. Hopefully it helped out, I had no better way of explaining it sorry. Try it out on a piece of paper with a pencil and ruler

@Adamin_The_III 3 ай бұрын

Finally, the proof of pythagorean theorem

@yimphockany808 4 ай бұрын

I like visual proof because it my favorite since I have ADHD

@andreashoppe1969 9 ай бұрын

Learned more in those 30 something seconds than in 8 years of school

@Mubashar783 9 ай бұрын

HY Dear pleases tell me, in which software it was created please tell me, I am curious to create like this one? pls sir😢

@machomancake 6 ай бұрын

What you showed there is that C^2 is equal to b^2 plus a PORTION of a ^2 and a wedge of the original triangle

@MathVisualProofs 6 ай бұрын

Follow the empty space, not the filled space. In the full diagram we start with c^2 empty space and end with a^2 plus b^2 empty space.

@unstableatom9922 9 ай бұрын

If geometry class was just visual representations of the math equations i feel more people would understand it

@Nitish_Kmr 9 ай бұрын

This is what our Vedas has already noted before Pythagoras.

@MMSatti 5 ай бұрын

C^2 has changed from the beginning of the video to the end. It was a square then it became a pentagon.

@jamesolatunji5 11 ай бұрын

The triangle is congruent between the a squared and b squared squares.

@coconutcute712 9 ай бұрын

“Shift and rotate” wouldn’t that make it… A SINGLE ROTATION?!

@slavoljubkostadinovic4299 8 ай бұрын

a2 + b2 + triangle = c2 + triangle => a2 + b2 = c2

@farisazim2941 5 ай бұрын

Now I know that Pythagora was an origami expert

@johndododoe1411 9 ай бұрын

There are other proofs that include WHY the pieces fit . Euclid's book explains how to prove all the rules from just 5 assumptions, which is why centuries of scientists used the word Geometry for anything proven with logic .

@mayukhbari 9 ай бұрын

This is technologies changing education system. Blackboard was ok but digital representation is easy to understand.

@marios_-nk4qg 8 ай бұрын

Or you can prove the Pythagorean Theorem by using trigonometry and say that c^2=2αb÷sin2a (the greek letter α represents the angle and not the side a)

@DanielPoupko 2 ай бұрын

Nice

@pietroaleo8371 8 ай бұрын

Thank you so much. Pietro italy, piedmont region,turin city, Grugliasco metropolitan area west of turin😮😅😊

@MathVisualProofs 8 ай бұрын

Glad to have you here!

@horstmueller1000 3 ай бұрын

Nope. It is true, that the a^2 + b^2 is uncovered . It is not clear and not visible at all, that the equation with c^2 is visible - there is a logical link missing. The area covered following the process consists of the original three shaded objects. Because the original is inside the original c^2 and all shaded forms are c^2 plus triangle, you can setup the equation of the theorem. And that is the missing link. In the narration. Not in the Visualization.

@pietroaleo8371 8 ай бұрын

Hallo! Do you have a graphic demo of carnot teorema? It is general form for generic triangles T😊🎉thank you a lot

@MathVisualProofs 8 ай бұрын

Not yet. I’ll think about it :)

@felixlafuente9714 11 ай бұрын

Shaded area equals a2+b2+1/2ab... Right?

@MathVisualProofs 11 ай бұрын

Yes. And after shifting is c2+1/2ab

@vk2ig 9 ай бұрын

I prefer Eddie Woo's explanation, as the end result is just the c² area (the square) composed of the a² and b² areas, and the original triangle isn't part of the area.

@MathVisualProofs 9 ай бұрын

Not sure which one you’re referring to exactly. I have over 10 visual proofs on my channel all citing sources. This one is one that was done in honor of the cut the knot creator who has passed away.

@averageday Ай бұрын

I’m confused there’s still an extra bit

@Miss.Princess8765 11 ай бұрын

Hay, I have a doubt..... The a^2 square that u slide down is filling the area of the original triangle So this pythagorean dissection is not appropriate as it is also taking the area of the original triangle, u r supposed to take the c^2 square for the measurement of side c....

@StormyBoi6 10 ай бұрын

The original triangle from c^2 is subtracted from a^2 to get a^2 + b^2 = c^2.

@japanlife_with_alfredo 6 ай бұрын

Good

@wensong4094 9 ай бұрын

Got it, Im on the bus and I cant hear anything. The blue triangle doesnt belong to the final area

@user-gu8qi8ye6r 7 ай бұрын

Такое доказательство более грамоздкое но всеравно спасибо❤

@mohammadrafhaan4092 8 ай бұрын

I have a doubt, aren't you including the area of the triangle as well for the area of square with c as it's side?

@MathVisualProofs 8 ай бұрын

This proof is about unshaded area in the entire diagram.

@mohammadrafhaan4092 8 ай бұрын

@@MathVisualProofs Sorry, My bad, I thought the unshaded area were the area of those two squares

@siddhimarketingmobilecompa9654 8 ай бұрын

Why did we use the purple triangle when a2 and b2 are equal to c2 so they should fill up the c2 the c2 square why please

@MathVisualProofs 8 ай бұрын

This is about the empty space in the diagram. The diagram includes all three squares and the triangle.

@OmnipresentPotato 9 ай бұрын

What about the original triangle? Why isn't its area considered?

@russelllim7177 Жыл бұрын

Thanks

@MathVisualProofs Жыл бұрын

Wow! Thank you so much!

@NicleT 10 ай бұрын

Cool, but I prefer the liquid example, much clearer, IMO.

@_PuffinMan 4 ай бұрын

What's the background music?

@wildocado5376 5 ай бұрын

how can this demostration be right id you also put the original triangle inside? and also a small piece of b² is out of the box.

@wensong4094 9 ай бұрын

I was expecting the fisrt square area plus the second squared are to shift and fit into the big square. Im too tired and wanted this to be straight forward . The colour dnare lols bigger than the big sqiared areaa.

@user-wj1qb3qu1y 9 ай бұрын

Does pythagoras theorem apply in any dimension? Or just in 2 dimension?

@deveshjha9116 7 ай бұрын

It is given in a hindu scripture Baudhāyana Śulbasûtra

@mlavarone8135 5 ай бұрын

I am not sure but , there is an error? c2=(a2+b2)-area triangle2 ? Can it be?

@guest6678 5 ай бұрын

And thus,a²+b²=c² ptherwise know as the *hypotonis*

@L8RT8RS 4 ай бұрын

IT MAY BE CLEVER, BUT WHAT ENDED UP BEING SHADED WAS A PENTAGON, NOT A SQUARE. THEY USED THE ORIGINAL TRIANGLE IN THE DEPICTION OF THE SQUARED SHADED PORTION. IT IS INACURATE.

@MathVisualProofs 4 ай бұрын

This is a proof based on the unshaded space. Follow the unshaded area

@Nol_B 9 ай бұрын

proof🔥🔥🔥

@Mubashar783 9 ай бұрын

HY Dear pleases tell me, in which software it was created please tell me, I am curious to create like this one? pls sir😢

@user-xq8zx9yt2r 8 ай бұрын

Can diameter area of c is equal to a or b if diameter is not equal in angel or shape so how things are equal every diameter has angel it's or its shape and size if angel and shape and size are equal so take aor b or c so then knowing equal

@zero_spin 11 ай бұрын

Hey wait.. why has the area of purple triangle been included?

@MathVisualProofs 11 ай бұрын

Just an alternate dissection proof. The idea is to look at the unshaded area in the full diagram. Since that triangle is present in both it has no effect on squares being equal.

@Rua9404 9 ай бұрын

i am now more confused than i am a minute ago

@Super_magi_c 9 ай бұрын

My god shapes

@buckleysangel7019 7 ай бұрын

1/phi squared + 1 squared = phi squared

@neilmorrone691 9 ай бұрын

Why are you using the AREA of the original BLUE TRIANGLE to fill the area of C x C square? Aren't you supposed to fill that C x C square ONLY using the areas of the other two squares A x A and B x B? So that A^2 + B^2 = C^2

@MathVisualProofs 9 ай бұрын

The proof uses the entire shown diagram. In this diagram (sometimes called the brides chair) we start with two unshaded squares with combined area of a^2 + b^2. After one cut and some rigid motions, we are left with an unshaded area of c^2 in the entire diagram. So these areas must be equal.

@MathVisualProofs 9 ай бұрын

Whoops. Backwards. Start with unshaded area of c^2 and end with two other squares unshaded. Still the idea is unshaded in the entire diagram.

@RoyalTortilla 6 ай бұрын

Or 1/2×a²+b²+(1/2×a×b) I think im wrong

@andytraiger4079 Жыл бұрын

In the last picture I can see that the square of a and the square of b have been moved down to occupy the square of c... but it didn't quite work, did it? The colored pieces are occupying the square of c, but they don't quite fit, do they; since the original triangle is occupied too, not just the square of c. There's that top of the red square that's sticking out of the c squared area. Why is that? It looks like the proof failed to me. The result of moving areas from a and b into c, should result in only the in area c being shaded, NOT having the original triangle shaded to! Can someone please explain why I'm thinking about this wrong?

@defaultdan7923 Жыл бұрын

i might be dumb but i feel like something is wrong in this explanation (the short). why is the original triangle in there when the equation makes no mention? idk, maybe someone can explain.

@MusaBinMasud-zu7us Жыл бұрын

The original triangle was occupied when c square was unshaded

@MusaBinMasud-zu7us Жыл бұрын

After that a square and b square and triangle occupied triangle and c square

@leif1075 Жыл бұрын

@@MusaBinMasud-zu7usyea the proof is wrong tge red swuare is sticking out of c..

@MusaBinMasud-zu7us Жыл бұрын

@@leif1075 ?

@spaceyeti7230 6 ай бұрын

I already knew this

@nikoraja4425 9 ай бұрын

Wait, isn't Pythagoras only for right triangles? Why does he use another triangle in his example? Can anyone explain?

@MathVisualProofs 9 ай бұрын

That is a right triangle and I say that at the beginning of the short. :)

@nikoraja4425 9 ай бұрын

@@MathVisualProofs i mean the visualization isn't right triangle, because there is not 90 degree, i was right? (I talked about visualization using the 3 squares)

@wyattstevens8574 5 ай бұрын

Neat!

@korianbeuty 11 ай бұрын

Can you explain it in a full video ,

@Sale_For_Alt 8 ай бұрын

Bro explained the Pythagorean theorem better than my math teacher

@eddiemanuelitonavarro2622 8 ай бұрын

But according to history it was the younger brother of phytagoras, astraynous who actually did the that equation

@JimmySchwietert 11 ай бұрын

Euclid's 47th problem

@fabiankrajewski3147 7 ай бұрын

a² + b² = c², it's that easy.

@kennethbransford820 9 ай бұрын

=== It is half of a circle ====

@Abcia2 6 ай бұрын

pretty sure there's an easier way to explain this but cool

@marcinkusiak9273 8 ай бұрын

a=3n b=4n c=5n

@Pythogoras570BC 9 ай бұрын

Does it matter that I don't get the procedure as proof? Similarly, you can fill the shaded areas with water and replace it into the unshaded area. This also proves the Pythagorean theorem but it's not a mathematical proof. Such steps remind me of the puzzles that can be solved by moving one stick to correct the equation.😢

@MathVisualProofs 9 ай бұрын

This one involves cutting and rigid motions so it constitutes a proof. The water one is a demonstration.

@Pythogoras570BC 9 ай бұрын

No need to argue fyrther. Thanks for replying, sorry for bothering@@MathVisualProofs