Origami Proof of the Pythagorean Theorem

Origami Proof of the Pythagorean Theorem


You don’t need numbers or fancy equations to prove the Pythagorean Theorem, all you need is a piece of paper. There is a ton of ways to prove it, and people are inventing new ones all the time, but I am going to show you my favorite. Only instead of looking at diagrams, we’re gonna fold it. First, you need a square, which you can probably obtain from a rectangle if you ask nicely. Step one, fold your square in half one way, then the other way, then across the diagonal. No need to make these creases sharp, we’re just taking advantage of the symetries of the square for the next step. But, be precise. Step 2: Make a crease along this triangle, parallel to the side of the triangle that has the edges of the paper. You can make it anywhere you want. This is where you are choosing how long and pointy, or short and fat, your right triangle is going to be, because this is a general proof. Now when you unwrap it, you’ll have a square centered in your square. Extend those creases and make them sharp, and now we’ve got we’ve got four lines all the same distance from the edges, which will allow us to make a bunch of right triangles that are all exactly the same. Step three: fold from this point to this one. Basically taking a diagonal of this rectangle. Now we’ve got our first right triangle. Which has the same shape and area as this one. Let’s call the sides: “A little leg”, “a big leg”, and “hypothenus”. Rotate ninety degrees, and fold back another triangle, which of course is just like the first. Repeat on the following two sides. The original paper minus those four triangles, gives us a lovely square. How much paper is this ? Well, the length of a side is the hypotenus of one of these triangles. So the area is the hypotenus squared. Step four: unfold, and this time let’s choose a different four triangles to fold back. Rip along one little leg, and fold back these two triangles. Then you can fold back another two over here. The area of the unfolded paper, minus four triangles, must be the same, no matter which four triangles you remove. So let’s see what we’ve got. We can divide this into two squares, This one has sides the length of the little leg of the triangle. And this one has sides as long as the big leg. So the area of both together, is little leg squared, plus big leg squared. Which has to be equal to this area, which is hypotenus squared. If you called the sides of your triangle something more abstract, like: a, b, and c, you’d of course have a squared plus b squared equals c squared So quick review: Step 0: Aquire a paper square. OK, Step One: Fold it in half three times. Step 2: Fold parallel to the edges anywhere you choose and extend the crease. Step Three: Fold back four right triangles around the square and admire the area hypothenus squared that is left over. Step Four: unfold and rip along a short side to fold back another four right triangles and admire the area one leg squared plus the other leg squared that is left. And that is all there is to it! Of course, mathematicians are rebels and never believe anything anyone tells them unless they can prove it for themselves. So be sure to not believe me when I tell you things like: This is a square. Think of a few ways you could convince yourself that no matter what the triangles on the outside look like, this will always be a square, and not some kind of a rombus or parallelogram or dolphin or something. Or, you know, maybe it is a dolphin, in which case you should define what a dolphin is and then show that this fits that definition. Also, these edges look like they line up together. Do they always do that? Is it exact?

100 thoughts on “Origami Proof of the Pythagorean Theorem

  1. If you guys are gonna keep this sillyness up I might as well just go and divide by zero, how would you like THAT!?

  2. a dolphin: something that is pale and can get wet without combustion. Paper=dolphin because dolphin =something that is pale and can get wet without combustion which also=paper. done. proof complete.

  3. A long time ago, I got a question involving the Pythagorean Theorem. I had no idea what it was, and later, I went through about five different dictionaries trying to find out what it was.

  4. you know if you divide the big side of an average piece of blank paper by the smaller one equals sqrt 2

  5. Unfortunately, it is not a proof. To make it into a proof, you have to do more than just fold the paper, you have to prove that this will work for any triangle, and that the paper must always fold into the shape we see it fold. She gets close to admitting this at 2:58–3:00, where she describes how mathematicians will complain about the 'proof' and suggests how to shore it up.

    But guess what! When I follow her suggestions, I still end up with a proof about as complicated as Euclid's famous proofs of the Pythagorean Theorem in Propositions 47 and 48 of Book I!

    Still, I have to admit it is interesting that there is an origami illustration of the theorem.

  6. I really enjoyed this video and it actually cleared some things up for me. Throughout my experience with Math I just accepted that the Pythagorean Theorem was the Pythagorean Theorem, and when my teachers tried to explain it to me through numbers I just pretended to understand it. This video actually visually shows why it works and it was super helpful! The only thing I didn't like about the video was that it went so quickly I had to replay it a few times to finally grasp the concept, but it is still very impressive. 

  7. This was a great video and interpretation of the Pythagorean Theorem! I really liked how you used a simple piece of paper to prove a theorem we use almost all the time in our every day lives and in school. Now I fully understand how the Pythagorean theorem works and have a totally new perspective on it. I also found it really funny!!

  8. I LOVE THE DOLPHIN COMMENT! Vi, you never cease to be entering, smart, funny, wonderful, and most importantly, ASOME!!!!:)

  9. It's pretty amazing how my online geometry class can make this lesson boring enough to fall asleep while reading. I even got excited about the heading including Pythagoras. But then I remembered your Youtube channel with actually exactly the same proof and it's interesting and fun and yes. 

  10. I learned you don't need numbers and equations to prove the pathagoreom theorem, you can prove it by just using a piece of paper.

  11. So I've been watching some of your videos. I like to watch at faster speeds now that youtube allows it (2x, 1.5x, etc). The thing I've found is that whether your video is complicated or simple, they are just hard to watch quickly. I mean this as a compliment. You explain super complicated subjects in what seems to me to be the least amount of time possible. Your videos are really well paced is what I am trying to say. If you read this. Thanks!

  12. Another way to find a triangular number is to plug it into the equation n(n+1)/2 or 1/2n^2 + 1/2n for short. In this case n is representing the number in the sequence you want to find. Here's two examples:
    12: 12(12+1)/2 -> 12(13)/2 -> 156/2 -> 78
    100: 100(100+1)/2 -> 100(101)/2 -> 10100/2 -> 5050

  13. Thanks for uploading this video. It helps my kids in the classroom understanding the Pythagorean Theorem. Bet your viewers will be increased because my students will watch this video for the project. Just consider it as a reward for your good deed! Thank you!
    PS: Who is here because of the Math project? For those who reply will get a bonus point lol

  14. Well I tried do this origami for a project (just to explain the stuff and all) and it doesnt freaking work!Wtf?

  15. Actually, a square is both a parallelogram and a rhombus, though whether it's a dolphin I know not.

  16. "Let's call them, little leg, big leg (me:aww cute) then hydornuxmdosm (me:o.O oh crap! What happened to the cuteness?!?!)"

  17. I do have complete proof that what you have called "paper" is indeed a dolphin, but I do not have the time, as I currently have to go perform some arbitrary task…

  18. türk çevirmene burdan teşekkür ediyorum sonunda türk bir çevirmen çıktıda anlayabildim sonunda videoyu 🙂

  19. What are you talking about?!? Of course, it isn't a square. Human beings as a species are so imprecise, that we can't even properly cut a straight line. No matter how close you believe you are, you will always be off by at least a little bit. And you ripped your paper, which means that each individual mini-jag in your rips will be considered an additional side. It is far closer to a centagon than a square. And there is also a high chance of your "square" being off by even the tiniest amount, resulting in a rectangle than a square. I mean, you never even said to fold that precisely.

  20. My favorite way is where there is water in squares on a triangle on a wheel, and the wheel rotates, so the water flows from a2 and b2 to c2

  21. They say the length of this video is pi and is not coincidence. But how can that be possible, vi hart said that pi = 4

  22. what is great is how this proof does not require 3 4 5 to solve, in some ways you could say it antedates all other proofs that would later follow, proofs still being conjured up to this day.
    what is great is how this particular proof is 'numberless' and might have been floating and folding around in the mind of creative man thousands of years before the fake Pythagoras was born and credited with this theorem. Which is an outright lie. He is NOT the source of this idea. At best he was a copycat who applied a new POV to a very old idea called the right angle.

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