# Visual Pythagorean demonstration

I know that there is a visual demonstration of $$a^2+b^2=c^2$$ using a smalĺ piece of paper, but there are also a lot of variations.

Which visual or drawing demonstration of the Pythagorean theorem can I show to 14-year-old students?

A number of proofs (43!) can be found at cut-the-knot.org. Some of these are described below.

I don't know how hands on you want the students to get with the visual aspect. I remember doing these proofs around that age. I recall this progression and it made sense to me why Pythagoras's Theorem was true.

1. Geometrical demonstration: Pythagoras's Theorem can be easily demonstrated. Construct squares on each side of the triangle, as shown below. Then, the large squares can be subdivided into smaller squares, which can be counted to be the same quantity. Also, I recall cutting the $c$ square to show that it overlaps exactly with the $a$ and $b$ squares. This technique can show it is true for particular values of $a,b,c$, but cannot prove the statement is true in general. (Image source: http://www.myastrologybook.com/PythagoreanTheorem16c.gif)

2. Pythagorean proof: This proof uses squares of equal sizes to show the desired relation. There is some manipulation of the squares required in this proof. (Image source: http://en.wikipedia.org/wiki/File:Pythagore.jpg)

3. Algebraic proof: This leads naturally from the second proof as a way of confirming what is being seen in the images. A detailed proof is given at mathsisfun.com. I will outline the basic steps below.

Each of the squares in Proof 2 has area $(a+b)^2$. In the left square, the total area is the area of the square plus the area of the four triangles, which is $c^2+4\times\frac12ab$. In the right square, the area is the sum of the two smaller squares plus the area of the for triangles, which is $a^2+b^2+4\times\frac12ab$. Equating both sides and simplifying gives the desired result.

• #1 is NOT a proof. If the third side were just a bit smaller or larger than what the Pythagorean theorem predicts, it would not be obvious. It is important to distinguish between proofs and ... demonstrations of meaning. – Sue VanHattum Mar 14 '14 at 0:22
• @SueVanHattum Perhaps "proof" isn't the right word then. Maybe demonstration is a better word. Given that, it is impossible to actually prove this using a visual method, because it will contain error (measurement by hand or pixel approximation by computer) in the drawing phase. – Daryl Mar 14 '14 at 1:00
• Number two is visual and is a proof. Since a demonstration was asked for, your answer would be great if you changed #1 to geometric demonstration. #2 is a geometric proof. – Sue VanHattum Mar 14 '14 at 1:05
• @SueVanHattum I have changed the description of #1 to demonstration, as I agree this is a better description. – Daryl Mar 14 '14 at 3:34
• It still mentions "these proofs" above, and "in this proofs" right after. It would be best to explain the difference. This is an example. "This technique can show it is true for particular values of a,b,c..." No, it doesn't even show that. Maybe it is close; maybe the hypotenuse is 4.9 not 5 on what looks like a 3-4-5 triangle. Showing the squares illustrates the idea, but doesn't show that it is true for even the one case. – Sue VanHattum Mar 14 '14 at 4:15

The best explanation I know comes from this answer by Emanuele Paolini (the only thing I did was to redo the pictures, please go and upvote Emanuele's post).

The point is that squares in the usual picture

doesn't have to be squares. The only thing is that we need to use the area (so that is scales with a square of the scale), it could have been pentagons.

But it could also be appropriate right triangles:

It is the similarity of the small triangles that makes it work (and explains why the triangle has to be right for the theorem to work).

I hope this helps $\ddot\smile$

• I really think this is "the" proof. – Steven Gubkin Dec 31 '15 at 21:39

Here is one of my favorite visualizations of the Pythagorean Theorem:

There is also a fun activity here. Have the kids draw two adjacent squares of any size of their choosing, and use the technique shown above to have them cut the squares into 5 pieces that can be arranged to form one larger square. The fun part about this is that the kids get do determine the discrepancy in the sizes of the initial squares, and no matter how large or small the discrepancy, this technique always works.

• Can you point to a description of how this activity works? – Sue VanHattum Mar 14 '14 at 0:31
• @SueVanHattum: This might be helpful. – Jared Mar 14 '14 at 0:42

Check out the book "Proofs Without Words" for a bunch of nice visual proofs of the Pythagorean Theorem.

Proofs Without Words: Exercises in Visual Thinking | Mathematical Association of America

• Didn't work for everybody… – MvG Apr 25 '14 at 18:41

• Could you provide a summary of the article so that if the link ever goes out of date, this answer still stands? You ought to be able to embed the link here. – Jon Ericson Mar 17 '14 at 5:23
• What I don't like about the water version is that it would be easy to similarly demonstrate false results. – Jessica B Dec 31 '15 at 8:34
• @JessicaB This is a visual demonstration of the meaning of the Pythagorean theorem. It is not a proof, but the author didn't ask for a visual demonstration of the proof. I think this is ideal for 14 year old students because it will grab their attention and show them what the theorem means. – Amy B Jan 1 '16 at 13:43
• Since Google+ has been shut down, the link to the posting no longer works. – user7990 Aug 30 '20 at 10:38

Maybe you can do President Garfield's proof and combine it with a history lesson.

page 161 of the New-England Journal of Education, April 1, 1876 (image from Google Books)

Note: M. C. = Member of Congress.

• This is my favorite proof, and the one that I tend to present to students. I like it not only for the historical significance, but also because I find the proof itself to be rather minimalist---there are just enough elements in Garfield's proof to push it through. – Xander Henderson Jul 3 '19 at 2:59

This was in my math folder, I don't know its source.

It was shown as 'not needing any further explanation', but of course, for lower level students, I connect the 3 points on the circle's perimeter to form the larger right triangle and remind them of the ratios formed by dropping an altitude (line b).

• Keeping in mind, that I found this, some time ago, and it struck me as very clever, what, if anything can I do to edit it into a demonstration that would have the same effect on others? Happy to edit to show how I present this live. It's a proof that doesn't assume the knowledge of pythagorean identity, only lower level geometry. – JTP - Apologise to Monica Jul 8 '19 at 14:17

Here is a Java applet showing Euclid's proof:

http://neil-strickland.staff.shef.ac.uk/courses/MAS100/pclock.html

• Please be sure to disclose your affiliation with the site you linked to. It would also help if you could summarize how the applet demonstrates the proof. – Jon Ericson Mar 17 '14 at 5:28
• My name is spelled out in full as part of the URL! – Neil Strickland Mar 17 '14 at 7:37
• Hi Neil, I get a blank screen when I click that link. Is the page still active? – JTP - Apologise to Monica Jul 3 '19 at 1:01
• No, browsers no longer support Java applets by default. I might redo it with different technology at some point. – Neil Strickland Jul 3 '19 at 1:11