# Kitchen Math: The Pythagorean Theorem Is Lurking Behind Your Sink

Maybe you hate the backsplash tiles behind your kitchen sink. I know I do. Not your tiles, I mean – mine. Their pattern is a boring grid; their color, a grubby off-white (if you’ve ever dug up grubs in your garden, you know what color I mean). A few of the tiles are stamped with chintzy brown sunflowers in a listless attempt at cheer. It probably doesn’t help that I hate our kitchen sink as well, a sink wedged in a countertop corner for added inconvenience.

But maybe you’re lucky. Maybe you have a convenient kitchen sink and exciting backsplash tile. If your backsplash tile is laid in a hopscotch pattern, you even have the Pythagorean theorem lurking right behind your sink. To the left is a backsplash much nicer than mine, tastefully tiled in a hopscotch pattern (well, tasteful aside from the peculiar choice of blue grout). To mathematicians, the hopscotch pattern is known as “Pythagorean tiling”, because of how beautifully it illustrates the Pythagorean theorem. Several proofs of the Pythagorean theorem exist, some of which are more intuitive than others. To me, the diagram superimposed on the hopscotch tiling produces one of the most intuitive proofs, especially for children. If you like, you can knock off reading right now and do the proof yourself: the diagram has all the labeling you need.

A right triangle is one containing a right angle (an angle measuring 90˚, or π/2 in radians). Its hypotenuse is the side of the triangle opposite the right angle. For the theorem, the length of the hypotenuse is called c, and the length of the triangle’s other two sides are called a and b.

The area of the square outlined in white is c^2. The area of the dark brown square inside it is (a – b)^2. The four right triangles surrounding the dark brown square each have area 1/2 ab. Because the dark brown square and the four triangles surrounding it add up to make the square outlined in white,

(a – b)^2 + 4 * 1/2 ab = c^2. Now expand (a – b)^2 and rewrite 4 * 1/2 as 2:

a^2 – 2ab + b^2 + 2ab = c^2. Notice 2ab and -2ab cancel:

a^2 + b^2 = c^2. Ta da!

That’s the Pythagorean theorem.

Simple. Kids only need to know how to expand a binomial in order to do this computation, something we learned at age 12, but which we could have been taught sooner. So why isn’t proving the Pythagorean theorem more widely taught?

I’m not sure. The same teacher who taught us binomials taught us the Pythagorean theorem and showed us a proof of it – the proof she happened to remember. Her proof was an asymmetric dissection – I don’t remember which one: the asymmetry made it harder for a kid to believe it would generalize to any right triangle, and very hard to remember the initial setup. A symmetric setup for proving the Pythagorean theorem, on the other hand, makes it easier to visualize how changing the lengths of a and b nonetheless leaves the formula a^2 + b^2 = c^2 (where c is the resulting hypotenuse) unchanged, and the symmetry also makes the setup memorable. I believe I was lucky, though, to have a teacher who bothered to show us any proof at all.

If your kitchen-sink backsplash is like mine, square tiles all of the same size in the typical boring grid, you can still use them to diagram a proof of the Pythagorean theorem, by nesting a square askew inside another square. That’s how Pythagoras did it. Follow the link and you’ll see a diagram different from the one above, but with the same symmetry. I happen to like the proof-by-hopscotch-tile, though, if only because I’m annoyed at my kitchen sink.

This post is the second in a series on kitchen math. The first one in the series is here.

1. Member
Quinn the Eskimo
@

It’s funny this should come up.  I’ve recently taken up an interest in re-learning some of my trigonometry.  Glad to know I am not the only person with right triangles on the brain.

2. Member
Matt Balzer
@MattBalzer

It’s always in the last place you look.

3. Contributor
@Midge

I know @percival has a proof of the Pythagorean theorem in his image cache.

If any of you have a favorite proof that is asymmetric, I’d be curious to hear why it’s your favorite.

4. Thatcher
Percival
@Percival

5. Thatcher
Percival
@Percival

6. Member
Randy Webster
@RandyWebster

As I mentioned earlier, I occasionally used trig when I was carpentering.  I used the Pythagorean theorem a lot. Six feet, eight feet and ten feet was the most common way to make sure you had a right angle.

7. Contributor
@Midge

anonymous (View Comment):
Very strictly speaking, don’t you need a lemma to prove the area of a right triangle with catheti a and b is ½ab?

If we’re doing everything from the bottom up, I should also include a demonstration that the expansion of (a – b)^2 really is a^2 – 2ab + b^2, like I said it was. The area of a right triangle is one of those things that can be understood, by a simple slice, as soon as a child is prepared to believe the area of an axb rectangle is ab. At least for me, knowing how to expand (a – b)^2 came considerably after I understood the area of a right triangle!

The various slice-and-dice arguments used to show the area of triangles – and, if you believe the circumference of a circle is 2π, the area of a circle, too – should be another kitchen math post. These are very simple ideas, and the kitchen conceit is hokey, but when I think back to the math I actually learned in elementary school, a lot of it was from stuff I read on my own that made the math similarly hokey – though I still maintain bananas are too bent to make slicing them a convincing demonstration of conic sections!

8. Member
RightAngles
@RightAngles

Randy Webster (View Comment):
Six feet, eight feet and ten feet was the most common way to make sure you had a right angle.

9. Member
Randy Webster
@RandyWebster

Randy Webster (View Comment):
Six feet, eight feet and ten feet was the most common way to make sure you had a right angle.

Lol.  I thought about saying @rightangles

10. Contributor
@Midge

President Garfield and the Pythagorean Theorem

I got to Garfield’s setup and paused to sketch it for myself. The darn thing is half the diagram used in Pythagoras’ original proof!

11. Contributor
@Midge

Randy Webster (View Comment):
As I mentioned earlier, I occasionally used trig when I was carpentering. I used the Pythagorean theorem a lot. Six feet, eight feet and ten feet was the most common way to make sure you had a right angle.

One of the rules I learned in engineering school is that in every problem you encountered, the triangle was always a right triangle and either 3–4–5 or 30°–60°–90°. In the real world, not so much.

We hardly needed the quadratic formula, either… funny, that…

12. Member
RightAngles
@RightAngles

I loved geometry in high school. Proofs were my favorite thing. I can’t remember how I did them anymore, I just remember liking it.

13. Member
RightAngles
@RightAngles

Randy Webster (View Comment):
Six feet, eight feet and ten feet was the most common way to make sure you had a right angle.

Lol. I thought about saying @rightangles

She said stop talking about her. So naturally, we can use these replies to keep talking about her.

You know me so well.

14. Contributor
@Midge

If any of you have a favorite proof that is asymmetric, I’d be curious to hear why it’s your favorite.

I can’t say it’s my favourite, because the symmetric proof in the original post is my favourite, but I have a certain fondness for Einstein’s boyhood proof because a) it was done by Einstein, and b) it is the first known instance where he employed a symmetry argument, which would be the basis of much of his work in adulthood. Here, the symmetry is more subtle because he used an invariant under scaling.

That is a fun kind of symmetry!

15. Contributor
@Midge

RightAngles (View Comment):
I loved geometry in high school. Proofs were my favorite thing. I can’t remember how I did them anymore, I just remember liking it.

In intro geometry I struggled with proofs a bit, because I’d make up theorems. Not intentionally – I just couldn’t keep straight which geometric relationships the class had already proven, and which, although they were intuitive, we hadn’t. Aside from that frustration – mostly frustrating because remembering which was which was the hardest part for me – I’ve always enjoyed proofs, and wish kids doing proofs for themselves was introduced into the curriculum sooner.

It’s a little deflating to be told, no, kids your age just aren’t developmentally ready to understand why you gotta use this or that formula. I’m sure some aren’t, but if you are, being told you aren’t and you wouldn’t understand only makes math scarier than it has to be.

16. Moderator

Kitchen calculation is actual a solid method for teaching the principle of arithmatic.

Give me M&Ms and ziploc bags, and I can do all grade school math.

17. Member
JustmeinAZ
@JustmeinAZ

RightAngles (View Comment):
I loved geometry in high school. Proofs were my favorite thing. I can’t remember how I did them anymore, I just remember liking it.

Me too! But it might have had something to do with my hunk of a geometry teacher. Oh, sorry, off topic.

18. Member
Mark Wilson
@MarkWilson

Randy Webster (View Comment):
As I mentioned earlier, I occasionally used trig when I was carpentering. I used the Pythagorean theorem a lot. Six feet, eight feet and ten feet was the most common way to make sure you had a right angle.

One of the rules I learned in engineering school is that in every problem you encountered, the triangle was always a right triangle and either 3–4–5 or 30°–60°–90°. In the real world, not so much.

We hardly needed the quadratic formula, either… funny, that…

Uh oh, the Ricochet Comment Temporal Paradox is back!

19. Member
Dean Murphy
@DeanMurphy

RightAngles (View Comment):
I loved geometry in high school. Proofs were my favorite thing. I can’t remember how I did them anymore, I just remember liking it.

Me too! But it might have had something to do with my hunk of a geometry teacher. Oh, sorry, off topic.

I liked geometric proofs too.  And later, I found writing programs to be so similar, I immediately took to it.

20. Contributor
@Midge

RightAngles (View Comment):
I loved geometry in high school. Proofs were my favorite thing. I can’t remember how I did them anymore, I just remember liking it.

Me too! But it might have had something to do with my hunk of a geometry teacher. Oh, sorry, off topic.

I liked geometric proofs too…

For the same reason? ;-P

21. Member
Dean Murphy
@DeanMurphy

RightAngles (View Comment):
I loved geometry in high school. Proofs were my favorite thing. I can’t remember how I did them anymore, I just remember liking it.

Me too! But it might have had something to do with my hunk of a geometry teacher. Oh, sorry, off topic.

I liked geometric proofs too…

For the same reason? ;-P

Probably not.  I don’t even remember the teacher.

22. Coolidge
JimGoneWild
@JimGoneWild

It’s always in the last place you look.

A strong kitchen cleanser and Brillo will get rid of it.

23. Member
Matt Bartle
@MattBartle

You people have this all wrong! The correct statement is

“The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”

I have it on good authority:

24. Thatcher
RushBabe49
@RushBabe49

Ewwww!!  Eek!!!!  Math!!!!!!!

25. Member
Randy Webster
@RandyWebster

“Math is hard.”–Barbie.

26. Contributor
@Midge

RushBabe49 (View Comment):
Ewwww!! Eek!!!! Math!!!!!!!

Don’t look now, but it’s lurking… It is always lurking…

27. Inactive
Anthea
@Anthea

I’ll be saving this post for the fall: my homeschooled son is heading into geometry/trig, and it’s been a long, long time, folks.

28. Contributor
@Midge

Anthea (View Comment):
I’ll be saving this post for the fall: my homeschooled son is heading into geometry/trig, and it’s been a long, long time, folks.

Trigonometric proofs are awesome! Or awesome when the Unit Circle is your friend (and considerably less awesome otherwise).

My Kitchen Math posts should continue to address geometric topics – I know we have at least one other mom here, Stina (@cm), who is a math mommy.

29. Thatcher
Percival
@Percival

anonymous (View Comment):
I’m curious, if there are any homeschooling parents among the readers here, what your opinion is of the merits of teaching plane geometry from the traditional approach of proof by Euclidean constructions based upon the five axioms versus diving early into Cartesian analytic geometry and trigonometry?

When I learned this stuff in the 1960s, we did about half a year on the ruler and compass stuff before learning about coordinate geometry. I felt cheated and betrayed: “I’ve spent a substantial fraction of my life (it is, when you’re a teenager) learning this ancient Greek stuff, and now, in one week, you’ve shown me how some Frenchmen got the answers out with about a twentieth the work by using algebra and arithmetic, which you taught me two years ago! (I felt the same thing when I discovered how many problems in mechanics just melted away when you applied conservation of energy.)

The argument was that doing the proofs trains the mind, just as they argued that learning Latin and Greek were valuable in understanding your own language and learning others, even if you didn’t plan to time travel to ancient Rome and Athens. But nobody doing modern mathematics does Euclidean proofs any more: they’re almost all doing algebraic proofs for which analytic geometry better prepares you.

Anybody want to go to bat for Euclid, for intrinsic and pedagogical value, as opposed to a recreation or appreciation of history?

I think it helps in connecting the concepts. Going straight to analytic geometry risks developing students who only know how to plug in numbers.

30. Moderator

Euclidean proofs are good for people with a knack for the visual, and provide a better relation to the actual figures.

It’s always important to teach the concept as well as the practice.  A lot of conservative critics of education dismiss this in favor of a pure focus on rote / practice.  The problem is that if you don’t understand what addition or area or a function or a derivative is, it will hurt your understanding.

Once I learned about vector field and the gradient, I finally understood what voltage was.

While we are talking math:

One thing I have been curious about for a while – how were sines, square roots, logarithms, etc. originally calculated?   I have no idea on how they were calculated by hand.

I’d also be curious if there are good books on using a slide rule, preferably with a slide rule to practice with.