The Pythagorean theorem is a celebrity: if an equation can make it into the Simpsons, I'd say it's well-known.

But most of us think the formula only applies to triangles and geometry. Think again. The Pythagorean Theorem can be used with **any shape** and for **any formula that squares a number**.

Read on to see how this 2500-year-old idea can help us understand computer science, physics, even the value of Web 2.0 social networks.

## Understanding How Area Works

I love seeing old topics in a new light and discovering the depth there. For example, I realize I didn't have a **deep grasp** of area until writing this article. Yes, we can rattle off equations, but do we really *understand* the nature of area? This fact may surprise you:

The area of **any shape** can be computed from **any line segment squared**. In a square, our "line segment" is usually a side, and the area is that side squared (side 5, area 25). In a circle, the line segment is often the radius, and the area is pi * r^2 (radius 5, area 25 pi). Easy enough.

**We can pick any line segment** and figure out area from it: every line segment has an "area factor" in this universal equation:

Shape |
Line Segment |
Area |
Area Factor |
---|---|---|---|

Square |
Side [s] | s^{2} |
1 |

Square |
Perimeter [p] | 1/16 p^{2} |
1/16 |

Square |
Diagonal [d] | 1/2 d^{2} |
1/2 |

Circle |
Radius [r] | pi r^{2} |
pi (3.14159...) |

For example, look at the diagonal of a square ("d"). A regular side is $\frac{d}{\sqrt{2}}$, so the area becomes $\frac{1}{2} d^2$. Our "area constant" is 1/2 in this case, if we want to use the diagonal as our line segment to be squared.

Now, use the **entire perimeter ("p")** as the line segment. A side is $\frac{p}{4}$, so the area is $\frac{p^2}{16}$. The area factor is 1/16 if we want to use $p^2$.

## Can we pick any line segment?

You bet. There is always **some** relationship between the "traditional" line segment (the side of a square), and the one you pick (the perimeter, which happens to be 4 times a side). Since we can convert between the "traditional" and "new" segment, it doesn't matter which one we use -- there'll just be a different area factor when we multiply it out.

## Can we pick any shape?

Sort of. A given area formula works for all **similar** shapes, where "similar" means "zoomed versions of each other". For example:

- All squares are similar (area always $s^2$)
- All circles are similar, too (area always $\pi r^2$)
- Triangles are
**not similar**: Some are fat and others skinny -- every "type" of triangle has its own area factor based on the line segment you are using. Change the shape of the triangle and the equation changes.

Yes, every triangle follows the rule "area = 1/2 base * height". But the relationship between base and height depends on the type of triangle (base = 2 * height, base = 3 * height, etc.), so even then the area factor will be different.

Why do we need similar shapes to keep the same area equation? Intuitively, when you zoom (scale) a shape, you're changing the absolute size but not the relative ratios within the shape. A square, no matter how zoomed, has a perimeter = 4 * side.

Because the "area factor" is based on ratios inside the shape, any shapes with the same "ratios" will follow the same formula. It's a bit like saying everyone's armspan is about equal to their height. No matter if you're a NBA basketball player or child, the equation holds because it's all relative. (This intuitive argument may not satisfy a mathematical mind -- in that case, take up your concerns with Euclid).

I hope these high-level concepts make sense:

- Area can be be found from
**any line segment squared**, not just the "side" or "radius" - Each line segment has a different "area factor"
- The same area equation works for similar shapes

## Intuitive Look at The Pythagorean Theorem

We can all agree the Pythagorean Theorem is true (here's 75 proofs). But most proofs offer a mechanical understanding: re-arrange the shapes, and voila, the equation holds. But is it really clear, intuitively, that it **must** be a^{2} + b^{2} = c^{2} and not 2a^{2} + b^{2} = c^{2}? No? Well, let's build some intuition.

There's one killer concept we need: **Any right triangle can be split into two similar right triangles.**

Cool, huh? Drawing a perpendicular line through the point splits a right triangle into two smaller ones. Geometry lovers, try the proof yourself: use angle-angle-angle similarity.

This diagram also makes something very clear:

- Area (Big) = Area (Medium) + Area (Small)

Makes sense, right? The smaller triangles were cut from the big one, so the areas must add up. And the kicker: because the **triangles are similar, they have the same area equation.**

Let's call the long side c (5), the middle side b (4), and the small side a (3). Our area equation for these triangles is:

where F is some area factor (6/25 or .24 in this case; the exact number doesn't matter). Now let's play with the equation:

Divide by F on both sides and you get:

Which is our famous theorem! You knew it was true, but now you **know why**:

- A triangle can be split into two smaller, similar ones
- Since the areas must add up, the squared hypotenuses (which determine area) must add up as well.

This takes a bit of time to see, but I hope the result is clear. How could the small triangles **not** add to the larger one?

Actually, it turns out the Pythagorean Theorem depends on the assumptions of Euclidean geometry and doesn't work on spheres or globes, for example. But we'll save that discussion for another time.

## Useful Application: Try Any Shape

We used triangles in our diagram, the simplest 2-D shape. But the line segment can belong to **any** shape. Take circles, for example:

Now what happens when we add them together?

You guessed it: Circle of radius 5 = Circle of radius 4 + Circle of radius 3.

Pretty wild, eh? We can multiply the Pythagorean Theorem by our area factor (pi, in this case) and come up with a relationship for any shape.

Remember, the line segment can be **any portion of the shape**. We could have picked the circle's radius, diameter, or circumference -- there would be a different area factor, but the 3-4-5 relationship would still hold.

So, whether you're adding up pizzas or Richard Nixon masks, the Pythagorean theorem helps you relate the areas of any similar shapes. Now that's something they didn't teach you in grade school.

## Useful Application: Conservation of Squares

The Pythagorean Theorem applies to **any** equation that has a square. The triangle-splitting means you can split any amount (c^{2}) into two smaller amounts (a^{2} + b^{2}) based on the sides of a right triangle. In reality, the "length" of a side can be distance, energy, work, time, or even people in a social network:

**Social Networks.**

Metcalfe's Law (if you believe it) says the value of a network is about n^{2} (the number of relationships). In terms of value,

- Network of 50M = Network of 40M + Network of 30M.

Pretty amazing -- the 2nd and 3rd networks have 70M people total, but they aren't a coherent whole. The network with 50 million people is as valuable as the others combined.

**Computer Science**

Some programs with n inputs take n^{2} time to run (bubble sort, for example). In terms of processing time:

- 50 inputs = 40 inputs + 30 inputs

Pretty interesting. 70 elements spread among two groups can be sorted as fast as 50 items in one group. (Yeah, there may be constant overhead/start up time, just work with me here).

Given this relationship, it makes sense to partition elements into separate groups and then sort the subgroups. Indeed, that's the approach used in quicksort, one of the best general-purpose sorting methods. The Pythagorean theorem helps show how sorting 50 combined elements can be as slow as sorting 30 and 40 separate ones.

**Surface Area**

The surface area of a sphere is 4 pi r^{2}. So, in terms of surface area of spheres:

- Area of radius 50 = area of radius 40 + area of radius 30

We don't often have spheres lying around, but boat hulls may have the same relationship (they're like deformed spheres, right?). Assuming the boats are similarly shaped, the paint needed to coat one 50 foot yacht could instead paint a 40 and 30-footer. Yowza.

**Physics**

If you remember your old physics classes, the kinetic energy of an object with mass m and velocity v is **1/2 m v ^{2}**. In terms of energy,

- Energy at 500 mph = Energy at 400 mph + Energy at 300 mph

With the energy used to accelerate one bullet to 500 mph, we could accelerate two others to 400 and 300 mph.

## Try Any Number

You can use any set of numbers that make a right triangle. For example, enter a total amount (50) and one subportion (30), and the remainder will appear below:

Suppose you want to see if a large pizza (16 inches) is bigger than two mediums (12 inches). Plug in 16 for C, and 12 for A. It looks like the large pizza can be split into a 12-inch and 10.5-inch pizza, so two-mediums are in fact bigger.

## Enjoy Your New Insight

Throughout our school life we think the Pythagorean Theorem is about triangles and geometry. **It's not.**

When you see a right triangle, realize the sides can represent the lengths of any portion of a shape, and the sides can represent variables in **any equation** that has a square. Maybe it's just me, but I find this pretty surprising.

There's much, much more to this beautiful theorem, such as measuring any distance. Enjoy.

Surprising Uses of the Pythagorean TheoremOctober 25, 2007 at 2:34 am[…] Turns out Pythagorean theorem isn’t only useful for triangle calculation only. This article explains it in details.read more | digg story […]

SpyrosOctober 25, 2007 at 4:44 amWow, thats really cool.

So if I’m getting this right, when a car is traveling at 100 mph, the energy that is required to take it to 110 mph (or the difference in its speed vector) is the same as that of two other cars travelling at 100 and 45,8 mph.

So, if they travel for 1 hour, they would have covered 100 miles (car A) and 110 miles (car C) respectively, but car A would have enough fuel to go for another 45 miles.

Makes you think about fuel economy/efficiency.. :)

HangOctober 10, 2017 at 9:01 amThat’s not true. Because the fuel energy is not only used to accelerate the car (from 0 to 100 mph), but also used to maintain the speed. Think about elastic collision of balls in vacuum.

KaizynOctober 25, 2007 at 8:42 amEinstein found this method of proof for Pythagorean Theorem when he was 12 or so; though he probably wasn’t the first (or last) to have done so.

KalidOctober 25, 2007 at 11:40 am@Spyros: I think that’s right from a pure energy point of view: the energy needed to go from 100-110mph equals the energy to go from 0-45.8 mph.

I’m not sure how well the pure energy analogy works (drag, efficiency of engine, etc.), but there is an ideal speed for fuel efficiency cars, and it’s not around 110mph :).

@Kaizyn: Thanks for the info. Yep, I like this particular proof compared to others because it focuses on the larger concepts, not just re-arrangement of area.

things to look at (October 24th - October 25th) | stimulantOctober 25, 2007 at 5:33 pm[…] Surprising Uses of the Pythagorean Theorem | BetterExplained […]

mccoynOctober 29, 2007 at 6:24 amSpyros,

Thats the energy required to accelerate the vehicle to that speed, not the energy required to maintain the speed, nearly all of which is required to overcome wind resistance at high speeds.

The coolest thing I’ve ever discovered about Pythagorean’s Theorem is an alternate way to calculate it. If you write a program that uses the distance form c = sqrt(a^2 + b^2) you will suffer from the lose of half of your available precision because the square root operation is last. A more accurate calculation is c = a * sqrt(1 + b^2 / a^2). If a is less than b, you should swap them and of course handle the special case of a = 0.

thak’s cool links » Surprising Uses of the Pythagorean TheoremOctober 29, 2007 at 8:08 am[…] Surprising Uses of the Pythagorean Theorem | BetterExplained. Whoa. This just completely blew my mind. […]

KalidOctober 29, 2007 at 8:26 amHi Mccoyn, thanks for the great info! The note about precision is especially useful, I hadn’t thought about the impact of order-of-operations on the calculation. I’m sure that comes in useful in graphics programming, etc.

jpsOctober 31, 2007 at 3:33 pmWow nice article. That puts the ‘ol Pythagorean theorem into a whole new perspective.

What program do you use to draw pictures in your articles?

KalidOctober 31, 2007 at 4:18 pmThanks, glad you liked the article. Yep, I’ve discovered that there are many gems to be found when we revisit concepts we “know” :).

I used PowerPoint 2007 to make the diagrams.

JDNovember 3, 2007 at 11:42 am“The Pythagorean theorem is a celebrity: if an equation can make it into the Simpsons, I’d say its well-known.”

Stopped reading right there.

KalidNovember 3, 2007 at 3:26 pmAh, that’s too bad — you may have found it interesting! [Both the article and the Simpsons episode :) ]

FjoolNovember 3, 2007 at 4:48 pmYour site is triggering a virus warning. Some java class it’s loading has a worm.

FjoolNovember 3, 2007 at 4:49 pmhttp://s3.amazonaws.com/instacalc/release/build516/javascripts/library-internal.js.packed

KalidNovember 3, 2007 at 10:22 pmHi, thanks for the info. Avast appears to be have a false positive (many AV companies mis-detect javascript as Feebs, see more here: http://isc.sans.org/diary.html?storyid=2319&rss).

I split my files into smaller chunks and it should be ok now. Appreciate the tip.

NotasDNovember 4, 2007 at 12:21 pmUsos sorprendentes del teorema de pitágoras…Haciendo esto

resulta que…

Measure Any Distance With The Pythagorean Theorem | BetterExplainedNovember 5, 2007 at 1:23 am[…] We’ve underestimated the Pythagorean theorem all along. It’s not about triangles; it can apply to any shape. It’s not about a, b and c; it applies to any formula with a squared term. […]

How to Develop a Mindset for Math | BetterExplainedNovember 27, 2007 at 2:31 am[…] The Pythagorean Theorem is not just about triangles. It is about the relationship between similar shapes, the distance between any set of numbers, and much more. […]

Per VognsenNovember 27, 2007 at 5:06 amYour proof of the Pythagorean theorem is very appealing. However, to be complete, you’d need to prove your supposition about similar triangle areas: concretely, the special case that the area of a right triangle can be computed as constant * hypotenuse^2. I tried a few different approaches, but they all end up having to apply Pythagoras in the end, which makes the whole exercise circular and thus ultimately pointless. If you try to establish the result by concrete computation it seems you are bound to run into this same difficulty–how do you get around it?

Per VognsenNovember 27, 2007 at 6:51 amIgnore that; I’m an idiot. If you treat it abstractly, it’s almost blindingly obvious.

You just have to prove that L^2 / A is constant within a similarity class. Take two members of the same similarity class, of areas A and A’ and lengths L and L’. Let F be the factor of the dilation that maps the first figure onto the other. Then A = F^2 * A’ and L = F * L. Squaring the length equation gives L^2 = F^2 * L’^2. Dividing the area equation by this, the F^2 factors cancel, yielding A / L^2 = A’ / L’^2. So the area to squared length ratio is indeed constant.

KalidNovember 27, 2007 at 10:10 amHi Per, glad you wrote it out :). No, you’ve saved me a bit of trouble as I wanted to discuss that aspect too!

RalphNovember 27, 2007 at 10:35 pmIf a “proof” of the Pythagorean Theorem does not bring in the Euclidean nature of the space under consideration, in what sense can it be considered a proof at all?

According to this wikipedia article, the “Euclidean metric” — which could also be called the “Pythagorean metric” — is actually one of the axioms of a properly-specified Euclidean geometry.

More interesting (to me, at least) would be a discussion of how the Pythagorean Metric leads to our concept of geometric area.

Sorry, I just can’t see how doing it the other way around makes sense.

KalidDecember 2, 2007 at 12:26 amHi Ralph, I think discussions of other geometries would be pretty interesting and a good follow-up (I need to read up on them). This wasn’t meant to be a rigorous proof from first principles, more a new intuition onto an old result.

harshilDecember 30, 2007 at 4:24 amHi

i m school kid

and dis seems to be terrific

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aaaaabbbbbOctober 2, 2017 at 5:04 amyou have said u will give the uses of pythagoras theorem but there is no uses at all

Mindset :: How to Develop a Mindset for Math :: February :: 2008February 5, 2008 at 3:09 am[…] The Pythagorean Theorem is not just about triangles. It is about the relationship between similar shapes, the distance between any set of numbers, and much more. […]

KalidFebruary 15, 2008 at 4:55 pmHi Harshil, glad you liked it.

Rescaling the Pythagorean Theorem | BetterExplainedFebruary 16, 2008 at 12:39 pm[…] The Pythagorean theorem can apply to any shape, not just triangles. It can measure nearly any type of distance. And yet this 2000-year-old formula is still showing us new tricks. […]

Wiskundemeisjes » De stelling van PythagorasMarch 15, 2008 at 7:47 am[…] Gelukkig is er BetterExplained, een site vol met betere verklaringen (en dan bedoel ik ook echt beter). In Surprising Uses of the Pythagorean Theorem geeft Khalid Azad een heldere uitleg. Elke rechthoekige driehoek kan in twee kleinere rechthoekige driehoeken verdeeld worden (in het plaatje is gekozen voor a = 3, b = 4 en c = 5, maar het geldt natuurlijk altijd). […]

AnonymousApril 29, 2008 at 5:10 pmi need a specific answer why circles are similar

Prehistoric Calculus: Discovering Pi | BetterExplainedMay 29, 2008 at 8:45 am[…] Inside square: side = sqrt(.52 + .52) = .7 [Thanks, Pythagoras], perimeter = 4 * .7 = 2.8 […]

abhinavJune 19, 2008 at 3:36 amhello, im a school student.. and i have a project on pythagorean theorem.. and i cant find out in concise the uses/applications of the same in daily life.. please help..

How To Measure Any Distance With The Pythagorean Theorem « Mytutorfriend- Your friendly tutorAugust 23, 2008 at 1:01 am[…] We’ve underestimated the Pythagorean theorem all along. It’s not about triangles; it can apply to any shape. It’s not about a, b and c; it applies to any formula with a squared term. […]

AnonymousSeptember 1, 2008 at 11:06 amhow does one calculate the height of i.e. a waterfall using the pathagorean

shashankSeptember 16, 2008 at 9:48 amexcellent , mind blowing work, i got a very big help from u for my maths project

KalidSeptember 16, 2008 at 6:22 pm@shashank: Glad it was helpful!

RyanSeptember 27, 2008 at 6:49 pmI’m doing a research paper and was wondering what your first and last name is so I can site the paper correctly. Thanks!

KalidSeptember 28, 2008 at 1:36 amHi Ryan, my name is Kalid Azad — good luck with your paper!

riteshOctober 15, 2008 at 3:43 amthis give an complete information on pythagoraus !

I now love maths more.

JohnOctober 17, 2008 at 8:41 pmKalid,

You might want to make it clear that you’re restricting your discussion to squares, since the statement

is not true for rectangles.

John MaxwellOctober 17, 2008 at 8:44 pmActually, I think I see what you mean. (Sorry, I’m commenting as I read. Maybe I should write down my comments and not say them until I’m finished reading, huh?)

John MaxwellOctober 17, 2008 at 9:26 pmI would like to see a derivation of mccoyn’s result of c = a * sqrt(1 + b^2 / a^2) being equivalent to c = a^2 + b^2, if anyone has one.

KalidOctober 18, 2008 at 3:24 pmHi John, thanks for the comments! No worries about the inline comments, it’s interesting seeing the thought process. Yep, rectangles can still follow the rule constant * side squared, but that constant will be different for each shape. In the case of a square, the constant is 1 (it is a different way to look at it).

The second result

c^2 = a^2 + b^2

c^2 = a^2 * (1 + b^2/a^2) [rearrange right side]

c = a * sqrt(1 + b^2/a^2) [square root of both sides]

Now, the physical meaning of this is interesting. It basically gives you a constant [sqrt(1 + b^2/a^2)] that maps you distance in the “a” direction to relative distance in the “c” direction.

There’s more details here if you like:

http://betterexplained.com/articles/rescaling-the-pythagorean-theorem/

KalidOctober 18, 2008 at 3:25 pm@ritesh: Glad you enjoyed it!

NareshNovember 2, 2008 at 3:09 pmThanks Kalid,Superb work,its crystal clear to understand.

KalidNovember 2, 2008 at 4:14 pmThanks Naresh!

Teresa WoodsNovember 10, 2008 at 2:43 pmwhy do we use the pythagorean theorem??? (plz answer)

KalidNovember 10, 2008 at 5:20 pm@Teresa: Hi, there’s several uses of the Pythagorean theorem mentioned in the article.

AnonymousFebruary 2, 2009 at 1:21 pmCan anyone explain the uses of Pythagorus theurom?(please can you answer this question)

AshleyFebruary 3, 2009 at 2:09 pmIn the section on proving the Pythagorean Theorem (Intuitive Look at the Pythagorean Theorem), I am unsure why the Area = F*hypotenuse^2. Can you please explain? I tried to use the A= 1/2bh equation and substitute one of the sides with h = sqrt(hyp^2 – B^2) but could not come back to the F * hyp^2.

Thanks,

Ashley

Lee'Anne ='(February 4, 2009 at 9:40 pmI Think This Is Good . But I Need Something That Could Be More/Better Explained !

KalidFebruary 11, 2009 at 7:21 am@Ashley: Hi, that equation

Area = F*hypotenuse^2

refers to the fact that any triangle can have an equation formula like this. The amount of F will change on the shape though.

Area = 1/2 b * h

is a more useful equation because it works for every type of triangle. But the first one gets at the idea that the area of any shape is essentially based on the hypotenuse squared (or any side squared, for a different F). For squares, area is 1 * side^2, or 1/16 * perimeter^2. In both cases, it’s “some number time a measurement squared”.

Hope this helps!

Better Explained « Xavier Seton’s BlogMay 7, 2009 at 12:38 am[…] Geometry: Pythagorean uses, Pythagorean distance, degrees vs radians […]

samarth singhMay 26, 2009 at 1:05 amthats all bakvass………

LAURAMay 28, 2009 at 4:23 amThank you very much ! It’s the best thing I’ve found about Pythagoras… :D

AnonymousJune 8, 2009 at 8:15 amCould u please explain me the concept of Area factor. Im really not getting it. With respect to various shapes how can i associate it with the given figure.

Sumit RaiJune 8, 2009 at 8:15 amCould u please explain me the concept of Area factor. Im really not getting it. With respect to various shapes how can i associate it with the given figure.

SudhirJune 25, 2009 at 7:37 amgot something new to know about pythagoras

george ducasJuly 23, 2009 at 3:28 pmhow can in incorporate Pythagoras into this theory

AmritAugust 1, 2009 at 10:22 amSorry, I am still gropling with the question – how the Pythagorean theory is helpful in our life?

KalidAugust 3, 2009 at 6:13 pm@Amrit: This is a tough question since the theorem is everywhere — it’s in the structure of the universe :). It’s a bit like asking how circles are useful in our life. Most people don’t “make” circles that often, but it’s a concept which is everywhere once you start looking for it.

If you ever need to find the distance between two things (driving, parts in a machine, diagrams on a piece of paper) the Pythagorean theorem was used. Most people don’t use it directly (most of us aren’t involved in computing things) but it’s one of the most useful results ever. Anything that involves a distance measurement likely involves the Pythagorean theorem; that’s just the starting point (it can define equations for circles, etc.).

DebAugust 13, 2009 at 5:12 amHi Kalid

Thanks for the well written article and great diagrams! I really enjoyed it and learned a lot. Your examples and humour were great and made for an enjoyable read – yes, Im learning that math can be enjoyable! :)

KalidAugust 13, 2009 at 10:02 am@Deb: You’re welcome, really glad you enjoyed it! I think any subject can be made enjoyable if looked at the right way :).

Of Not Missing “The Point” | Furkan CALISKANAugust 14, 2009 at 5:24 am[…] [1] : http://betterexplained.com/articles/surprising-uses-of-the-pythagorean-theorem/ [2] : http://en.wikipedia.org/wiki/Pythagorean_theorem Share and Enjoy: […]

sravanthiNovember 23, 2009 at 6:32 amcan v use pythagoros theorem in the field of medicine. ( please answer )

MarkJanuary 14, 2010 at 7:24 amI really like the boat thing. You can also put it this way: if with a bucket of paint I can paint the hull of an “a” feet boat, how long a boat can I paint (assuming the boats are similar) if I have 2 buckets? Answer: I can paint a boat which is a*Sqrt(2) long (and not twice as long!).

More generally, if with one bucket of paint I can do an “a” long boat, with “n” buckets of paint, I can do a boat which is x=a*Sqrt(n) long. That’s because k*x^2=k*n*a^2, where k is the form coefficient: the two “k” cancel out, and we solve for “x”, discarding, of course, the negative solution. This “k” thing is the reason why Pythagorean theorem holds for any shape: “k” cancels out when we solve.

But what if we want to paint many small boats with the same bucket that allows us to do an “l” long boat? As you stated, any triangle can be split in 2 similar triangles. But then, any form, e.g. the hull of a boat, can be split in 2 similar forms and so on and on. Let the surface of the boat be k*l^2. We can paint as many smaller boats as we want (all the same or different), as long as k*l^2=k*a^2+k*b^2+k^c^2… or, after “k” cancels out, l^2=a^2+b^2+c^2…

If the smaller boats are all the same, and we have “n” of them, that becomes l^2=n*x^2 and, if we solve for x, x=(l^2)/(Sqrt(n)), discarding the negative solution.

If feel that Pythagorean theorem can indeed be extanded beyond triangles, but then, we are left with plain algebra to work with. Pythagorean theorem also can’t easily solve the “n” boats problem: we must use algebra for that.

AnonymousMarch 2, 2010 at 2:30 pmWOW

antoniaMay 18, 2010 at 1:10 amLOVE it!

wow you have helped me sooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo(etc.)

much on my project!

thhanks, i’m impressed.:)

KalidMay 18, 2010 at 9:35 pm@antonia: Awesome, glad you liked it!

YASHMay 22, 2010 at 11:45 pmyou really helped out in my school project. thanks!

KalidMay 24, 2010 at 5:49 pm@Yash: You’re welcome!

AnonymousMay 30, 2010 at 12:34 amThis dint help at all :(

KalidMay 31, 2010 at 10:33 pm@Anon: Sorry it didn’t help, feel free to leave a comment about what parts were confusing.

SwatiJune 3, 2010 at 11:40 pmIts good! I always find maths difficult to understand so this too was a bit difficult.

Thanks. My maths project has become easy.

KalidJune 6, 2010 at 11:33 pm@Swati: Thanks, glad it helped!

MollieJune 14, 2010 at 1:51 ami have a maths question : investigate wheather the theorem holds if equilaterial triangkes are drawn instead of squares?

Ed WattsJuly 16, 2010 at 6:00 amQuick one on the SQRT computer formulation of

Pythagoras – “If you write a program that uses the distance form c = sqrt(a^2 + b^2) you will suffer from the lose of half of your available precision because the square root operation is last. A more accurate calculation is c = a * sqrt(1 + b^2 / a^2). If a is less than b, you should swap them and of course handle the special case of a = 0.

mccoyn — October 29, 2007 @ 6:24 am

“

I can follow the logic with one exception the ‘discovery’ that where a<b that the formula requires b to be switched for a. Its true as

1 + a^2/b^2 does not equal 1 + b^2/a^2 numerically

but what defines which of the relevant 'sides'

in the original theorem should come first since

any order in their squaring will hold for

a^2 + b^2 = c^2 ….. thanks

chandanAugust 20, 2010 at 2:39 amhey..your articles are simply superb..while reading this one i started thinking as to what s the intuition behind area of a square being side^2 etc..and then came the “aha”moment..if we place these line segments one above the other there will be total of s(side) line segments..so the total area is side*area of one line segment..since lines are having unit thickness area of 1 line segment is simply its length..and thus area of square=side^2..

in fact if we consider the diagonal of a square, we can move it to the corners and then we span a square twice the size of the original square..so actual area=1/2 * d^2..

same way we can think of other areas too..its amazing!!!

chandanAugust 20, 2010 at 2:41 amand one more point..since we can wrap any shape in a square there is always going to be a (line segment)^2 in the area..

chandanAugust 20, 2010 at 2:46 amone more observation..for a circle, if we find the best enclosing square, the total area of that square is 4*(radius^2)..but the actual area has to be lesser than this..so we have pi instead of 4.but can you tell me how this pi was reached??

AnonymousSeptember 1, 2010 at 1:33 pmdarn you, Kalid! i’m in 3rd year calculus, but now all I want to do is go back to geometry again!

seriously, thanks for putting this up. this makes math fun again.

KalidSeptember 13, 2010 at 4:33 pm@Anonymous: Awesome, glad it helped! Yeah, it’s so easy to get distracted when math becomes enjoyable again :).

Meme-ingful Education | Blog | design mindSeptember 22, 2010 at 9:46 am[…] And while she emphasized learning how to think, perhaps if she had told her students they could calculate the power of their social networks, she would have seen a renewed appreciation for an ancient […]

What are some neat visual proofs of Pythagoras' theorem? - QuoraOctober 20, 2010 at 12:10 pm[…] Here's an illustration, with credit going to (and further explanation found at) the site http://betterexplained.com/artic…. Note that the blue triangle's top-left angle makes it similar to the green triangle, and […]

Like, part 2 « ArcsecondOctober 30, 2010 at 1:53 am[…] continued, reading an article on the Pythagorean theorem, looking for something to rip apart. On finishing, I felt a deflated. It […]

EyvindNovember 2, 2010 at 4:24 amFor those of you that wonders if if maths are useful. Understandig Pythagoras helped me building my home. How to make 90 degrees corners in a house? Knowing Pythagoras says that marking 3 feets from the corner of one wall and 4 feets of the adjacent wall and then using the marks adjusting the sides till you get your 5 feet diagonal and voila.. the sides forms a perfect 90 degree angle.

KalidDecember 7, 2010 at 11:14 pm@Eyvind: That’s a great example of going backwards — yes, you can make a perfect right angle at home by making a triangle of sides 3-4-5!

johnnyJanuary 10, 2011 at 1:47 pmWow those are great examples. I found some other site that actually has circle, triangle, rectangle, and many other calculators.

http://www.thexorb.com/Algebra/Pythagorean/Pythagorean-Theorem-Calculation.aspx

AndrewJanuary 20, 2011 at 9:47 amI was confused by the Area Factor bit as well. So I did a little formulating.

Starting with: Area = F * hypotenuse^2

Replace Area: 1/2 Base * Height = F * hypotenuse^2

Rearrange: (Base * Height) / (2 * hypotenuse^2) = F

There is your area factor.

Making this useful, if you only knew you had two _similar_ triangles with hypotenuses 3 and 4 respectively (as in the example) simply go back to Pythagoreas and solve for the missing hypotenuse of the combined triangles, knowing of course that the respective smaller hypotenuses become the sides of the larger triangle.

As Kalid said the Area Factor doesn’t really matter, but I still wanted to find a formula for it…voila.

rober van dijkMarch 23, 2011 at 11:40 amvision sort of locked 3d sighting “Topologie ia branche mat who you presenting ability see it plus de sides is the key of a triangles example bending Arcsin(x):

sides is statement the many mat formats

krob

YaakovMarch 24, 2011 at 2:47 pmBrilliant. Exactly what I was looking for.

tahiraApril 21, 2011 at 10:58 amthis was so interesting……..

tahiraApril 21, 2011 at 11:07 amcan’t be better than this

KalidApril 21, 2011 at 11:26 pm@tahira: Thanks!

hankaMay 14, 2011 at 9:21 pmThanks. Your explanations on the physic were brilliant. It must surely also applies to others like the

mgh,pVg(buoyant force),hpg(pressure)and others.KalidMay 17, 2011 at 7:28 am@Hanka: Thank you!

caesarMay 26, 2011 at 5:40 amI would like to add to the application to sorting algorithms. Yes, it is easier to partition the input and sort them separately. But where has the remaining time/effort gone? It is in order to “merge” the two partitioned sets into a single sorted array. However this takes O(n) and not O(n^2). This is an early indication that sorting is possible in lesser than O(n^2) time.

NANDEESHAugust 14, 2011 at 8:55 pmHi,

Everyday application of Pythogoras theorem(extended to power 3)is in buying coconuts/fruits.

If a fruit of 3″dia costs 10 bucks, a fruit of 4″ dia costing less than 23 bucks as well as a fruit of 5″dia costing less than 46 bucks are cheaper.

KalidAugust 14, 2011 at 9:47 pm@NANDEESH: Ah, in this case I think it might be more of a regular area comparison?

3″ dia = 1.5″ rad = 2.25 * pi area

4″ dia = 2″ rad = 4 * pi area. As long as it’s cheaper than 10 * (4/2.25) = 17.77 it will be better deal.

NANDEESHAugust 15, 2011 at 1:46 amI suppose we have to compare volumes rather than areas of cross section.

More precisely, we may have to compare shell volumes (of say 1″ thickness) in case of coconuts.

KalidAugust 18, 2011 at 9:13 pm@NANDEESH: Ah! Yes, that makes sense — my mistake! :)

--billSeptember 26, 2011 at 6:17 amHow does this work for areas under the graph of y = x^2?

The Great Equations » Ishango BonesOctober 5, 2011 at 11:21 am[…] Here are some surprising uses of the Pythagorean Theorem […]

SilviuOctober 27, 2011 at 3:30 amThe Pythagorean theorem is based on *similarity*. I found a quite interesting site at: http://www.echteinfach.tv/2011/10/rechtwinklige-dreiecke-satz-pythagoras.html#w

If you look for the paragraph “Das Geheimnis hinter dem Satz des Pythagoras” (the secret behind Pythagoras) you find some interesting stuff:

Conclusion: All 3 triangles are similar, so all 3 squares are similar. This is the foundation of the theorem.

The description that I like as well (‘expanded similarity’?): each square originates from its corresponding triangle, it’s the triangle itself, increased in size, and changed in form! See images on the website to understand that.

CrazyFeetKait(:December 2, 2011 at 7:33 amWoah. Simply mind blowing from what can derive from one simple little theory!

Who knew the Pythagorean Theorum had so much more potential that many didn’t even know about! :o

The only part that had me a bit confuzzled was:

“For example, look at the diagonal of a square (“d”). A regular side is d/sqrt(2), so the area becomes 1/2 d2. Our “area contant” is 1/2 in this case, if we want to use the diagonal as our line segment to be squared.

Now, use the entire perimeter (“p”) as the line segment. A side is p/4, so the area is p2/16. The area factor is 1/16 if we want to use p2.”

The whole area factor concept is hard to catch onto…

(the area becomes 1/2 d2)

(A side is p/4, so the area is p2/16. The area factor is 1/16 if we want to use p2.”)

But overall, this entire article is Mathtastic! :D

kalidDecember 2, 2011 at 11:54 am@CrazyFeetKait: Awesome, glad you enjoyed it! Thanks for the feedback, I would like to clarify that part, probably with a diagram.

I see the area factor as the conversion rate depending on what you want to measure. One way to think about it:

* The average person has their armspan equal to their height (“height factor of 1”)

* The average person has their height equal to 10 times their foot size (“height factor of 10”)

I’m not sure if the 10x is exactly right, but the idea is that depending on what part you measure, you get a different “formula” (armspan * 1, foot size * 10, etc.). The neat thing is the formula works on babies to adults since it’s in term of “armspan” and not absolute height. Hope this helps a bit! I would like to clarify this more.

sabrinaJanuary 7, 2012 at 8:22 pmi like this website it help me finish a math report i will definetly use this website more often there is so much i learned just on the first page i didnt believe that you could uses for any shape that can be squared so i tried it out and its true you can that just made math so much easier

kalidJanuary 9, 2012 at 1:31 pm@sabrina: Thanks, glad you liked it!

patrickFebruary 14, 2012 at 10:00 pmthat was the most fantastic idea i ever came across in maths.

kalidFebruary 15, 2012 at 9:28 am@patrick: Thanks!

tieraMay 10, 2012 at 7:40 amlove this

Mathematics: Which is the most beautiful proof of the Pythagoras theorem? - QuoraJuly 6, 2012 at 2:17 am[…] Post • 12:49am Anders Kaseorg, MIT ’08, S.B. in Mathematics with Com…This one.http://betterexplained.com/artic…… (more) Sign up for free to read the full text. Login if you already have an account.Comment […]

Math As Language: Understanding the Equals Sign | BetterExplainedSeptember 25, 2012 at 1:30 pm[…] a mathematical truth like a^2 + b^2 = c^2, were a, b, and c are the sides of a right […]

MaryOctober 31, 2012 at 4:05 pmI can’t seem to understand this:

“Assuming the boats are similarly shaped, the paint needed to coat one 50 foot yacht could instead paint a 40 and 30-footer.”

I don’t understand how physically that’s possible, or the fact that two separate things could equal larger area, but end up equaling a smaller one as the hypotenuse. I’m sorry :( I’ve read over this a few times over the course of a few days but it’s not clicking. Mark’s comment does not help. (jan 14th 2010)

Ian MaxwellSeptember 19, 2018 at 10:04 amThis response is coming very late, but maybe it will help. The yachts being discussed have _lengths_ of 30, 40, and 50 feet respectively. Their _areas_ are then proportional to 30^2, 40^2, and 50^2 respectively — that is 900, 1600, and 2500. And 900 + 1600 = 2500. (This assumes that the yachts are otherwise proportioned the same, so that the longer ones are also wider and taller — if they’re *just* wider then this doesn’t work.)

By way of analogy, consider a perfectly cubic room with the floor and ceiling and all the walls covered in equally-sized tiles. If you want to tile two such rooms of the same size, you will only need twice as many tiles as you would need for one. But if you want to tile a single room of twice the height (and therefore twice the length, etc.) you will need four times as many tiles. This is because the number you need doesn’t scale with the linear dimension but with the square thereof.

Now replace ‘room’ with ‘yacht’, and ’tiles’ with ‘paint’, and maybe you have a better idea of what’s happening here.

Why They Teach You Math // Moment Matters | Interesting articles you can relateDecember 10, 2012 at 10:31 am[…] also takes credit on measurements. For instance, you can apply the pythagorean theorem to know the shortest way to go to one place. Even angles are given too much focus on this area; so […]

KalidJune 28, 2013 at 10:04 pmHi Mary, the idea is that a yacht is essentially a deformed sphere. Spheres have surface area 4 * pi * r^2.

So we can do

50^2 = 40^2 + 30^2

and multiply both sides by 4 * pi to get

4*pi*50^2 = 4*pi*40^2 + 4*pi*30^2

which says “A ball of radius 50 has the same surface area as a ball of radius 40 and a ball of radius 30”. In our case, we’re using ship hulls, vs. perfect spheres, but the relationship is (theoretically) the same.

anirudh mathurJuly 6, 2013 at 8:10 amman this is awesume. it has helped me a lot in my project

RodAugust 4, 2013 at 6:33 amThe Pythagorean Theorem as modern art, featuring a squared circle:

http://www.aitnaru.org/images/Pi_Corral.pdf (see Pythagorean Pi design)

Geometers easily comprehend that this new concept of Pi simply complements one ratio (Pi) with another (ASR) … and both ratios include the same mysterious and stimulating essence of irrationality! Such is the nature of squared circles.

How not to square the circle?

Believe that it is impossible.

“Sacred” Geometry Theory of Everything | World Mysteries TVAugust 18, 2013 at 2:26 pm[…] “Pythagorean” Theorem. (Scholars suggest Pythagoras learned it in present day India)http://betterexplained.com/articles/surprising-uses-of-the-pythagorean-theorem/ […]

"Sacred" Geometry Theory of Everything - World MysteriesTV | World MysteriesTVJanuary 19, 2014 at 12:11 am[…] “Pythagorean” Theorem. (Scholars suggest Pythagoras learned it in present day India)http://betterexplained.com/articles/surprising-uses-of-the-pythagorean-theorem/ […]

How To Learn Trigonometry Intuitively | BetterExplainedFebruary 24, 2014 at 8:30 am[…] the Pythagorean Theorem (a2 + b2 = c2) we see how the sides of each triangle are […]

Adding Apples and Oranges in Multiple Dimensions | Zach's HacksMarch 15, 2014 at 10:50 pm[…] Pythagorean theorem is the single most important math equation you’ll ever encounter. At it’s most obvious it […]

JustinApril 15, 2014 at 11:21 amJust curious on your mention of using perimeter to get area on a square. When you use the formula Area = (1/16)P^2 , there is no way that makes sense. Simply using a 10×10 square, you can figure out this equation doesn’t work out when you wind up with a (100/16) answer instead of 100. What is this?

kalidApril 15, 2014 at 12:43 pmHi Justin, great question. In the case of a 10×10 we have Perimeter = 40. So we can plug in P = 40 and get:

Area = (1/16) * (40)^2 = (1/16) * 1600 = 100

Hope that helps!

maxJune 2, 2014 at 4:07 amthis website is no help at all!!!!!!!!!!!!!!!

sahilJune 19, 2014 at 9:57 pmi have got my work from this amazing site

Willie JohnsonJune 27, 2014 at 2:32 pmGreat work!! And a big thank you for confirming some of my own theories . I pretty much used the same reasoning to prove several mathematical conjectures such as Fermat’s Last Theorem. I know. Wiles has already proved it but certainly not like what Fermat would have. Upon seeing Fermat’s equation I immediately said that’s just the Pythagorean theorem. Its almost exactly like you said except the “factor” is N^n-2. But since you have a,b, and c then N^n-2 is a different factor for each variable. But you can get around that by expressing all the variables in terms of c or the hypoteneuse, i.e., a=c*sine and b =c*cos. Gives a very short sweet margin proof more in keeping with what Fermat actually came up with.

I am convinced that if mathematician would actually sit down and think about some of these problems intuitively rather than trying to impress other mathematicians with “rigor” then far more long standing mathematical problems would be solved.

Learning math? Think like a cartoonist. | BetterExplainedJuly 24, 2014 at 12:47 pm[…] The Pythagorean Theorem explains how all shapes behave (not just triangles). Don’t whip out a geometric proof specific to triangles. See what circles, squares, and triangles have in common, and show that the idea works for any shape. […]

David CaryJuly 28, 2014 at 11:12 pmIt turns out that

is true even when area is measured by some other shape — when area is measured by something other than tiny squares.

http://math.stackexchange.com/questions/800699/why-square-units

gaurangaJuly 29, 2014 at 6:22 ami liked it

AnonymousSeptember 8, 2014 at 2:23 amwow .thanks men you help me in our science ka look alike

AnonymousSeptember 9, 2014 at 9:00 amMaybe I’m just being slow, but I don’t see much more than just manipulation of equations here! It’s not incredibly profound.

nikoSeptember 26, 2014 at 3:40 amhi Kalid hope you have time:

so if i see c as a vector that can be split into components (a,b) i can just as well see the area defined by c as a vector that can be split into components of the areas defined by a and b.

so when we split the right triangle into two self-similar triangles, we really split the area into the components of its growth in the two orthogonal dimensions…

what do you think of this? i found this really beautiful but i just wonder how to think of it.

then what we see as the area is really the magnitude of the area vector just as the hypotenuse is just the magnitude of the vector c. and what is magnitude? in fact it is just projection in a different coordinate system..

somehow this is really whirling up quite a bit of my understanding of direction. pretty interesting..

nikoSeptember 27, 2014 at 3:40 amthe comment might have become a bit confusing in the end.

i guess what im really find interesting is that in the right triangle the self similar areas defined by the two sides add up to the area defined by the hypotenuse, such that one might say that the area is split into the two dimensional components, but the lenghts of the sides obviously dont add up taxicab-like to the lenght of the hypotenuse. yet in the end there is this beautiful relationship of the area to the euclidean norm/the hypotenuse.

well in the end i think that i should probably stay humble and let those patterns be as awesome as they are, instead of speculating all too much. sure its nice to think about those patterns, but i just looked at a pair and it was of such beautiful complex structure, that i just felt like i should spend my time rather on using the patterns to construct something nice. the patterns are there anyway and for the pythagorean theorem i actually have quite a nice intuition already, thanks to your article :)

Phil JilbertOctober 3, 2014 at 10:39 amHi Khalid, I noticed that you applied the relationship for the well known 5×5 = 4×4 + 3×3 Pythagorean to circular areas. Did you know that 6x6x6 = 5x5x5 + 4x4x4 + 3x3x3 would similarly apply to spherical volumes?

RodOctober 30, 2014 at 7:03 pmA three-points geometric model of Pi?

Consider this remarkable Pythagorean triangle that defines both a circle and its square. The center of the circle is located at the point where the perpendicular to the hypotenuse connects with the triangle’s long side; circle’s diameter = 4:

Long side = Pi = 3.14159265358979323846264..

Hypotenuse = (square root of Pi) x 2

= 3.54490770181103205459633..

= side length of circle’s square.

The related circle-squaring inscribed scalene triangle is inherent in this Pythagorean geometry. The scalene’s short side has length equal to one side of a square inscribed in the circle; the longest side forms a 45-degree angle with the hypotenuse of the right triangle.

Intuition For The Law Of Cosines | BetterExplainedNovember 13, 2014 at 4:16 pm[…] Law of Cosines resembles the Pythagorean Theorem, […]

NicolayNovember 20, 2014 at 11:17 pmHei Kalid!

Another great article, man! One question: How does one intuitively infer that “A regular side is d/sqrt(2)” where d is the diagonal?

NicolayNovember 20, 2014 at 11:27 pmOkay, I just realised the answer to my question – which in retrospect I should’ve inferred from the title of the post.

For anyone wondering:

d = hypotenuse

s = the length of the sides

So we have:

d^2 = s^2 + s^2

d^2 = 2s^2

1/2 * d^2 = s^2

d/sqrt(2) = s

… I guess it was kinda obvious. :p

kalidNovember 21, 2014 at 9:35 amHi Nicolay, great question. It’s figure-out-able after some work but not “obvious” (in that you should see it immediately). Thanks for working through it, it’ll help people!

YanaMarch 7, 2015 at 3:22 amWow, that blew my mind! I’d always thought the Pythagorean Theorem was kind of dull, but you gave it a whole new twist. :) And the whole Area=factor*(line segment)^2 was also pretty amazing. I’d never thought of it that way before. Keep up the good work.

kalidMarch 8, 2015 at 11:17 pmThanks Yana, glad you enjoyed it!

Scarecrows, Pythagoras, and Theorems OH MY! | Online Learning TipsMarch 13, 2015 at 10:31 am[…] Kalid, founder of BetterExplained. Surprising Uses of the Pythagorean Theorem. […]

JackApril 8, 2015 at 10:03 amI think the reason the kinetic energy theorem works is this: The force of an object is mass times acceleration, you square the velocity and multiply by 0.5 because acceleration is measured in squares and velocity is the dirivive of acceleration so to balance out the equation you divide by 2.

Lavanya.sApril 20, 2015 at 2:30 ami dont know what to say its helped me a lot

ShreyaMay 6, 2015 at 3:27 amI find difficult 2 find the uses of the Pythagoras theorems extension………can sum 1 plzz help me find it…… Plzz…..

Nayiga JaneMay 9, 2015 at 2:04 amAdvantages of using pythagorean theorem are not discussed in the notes above.

RheanMay 18, 2015 at 4:42 amhi Kalid, I’m a new math teacher. Please recommend a good project about Pythagorean theorem for my students to do.

RheanMay 18, 2015 at 4:50 amThank you so much.

karanMay 24, 2015 at 12:08 amHow can you write 5 =4+3 ???

Tim Shelton-JonesJune 5, 2015 at 1:56 pmAwesome website, a dose of mathematical sanity, this is the real thing – mathematical meaning before mathematical manipulation (though sometimes we seem to need to learn the manipulation first). (By the way Karan (No. 122), Kalid is not saying “5 = 4 + 3”, but “the area of a circle radius 5 = area of a circle radius 4 + area of a circle radius 3”.)

not uuuuuuuuuuuuuuseful {sorry}June 27, 2015 at 1:29 amnot useful at all

Mohammed aliSeptember 22, 2015 at 9:05 pmTwo Mango’s of 300 gram and 400 gram equal to One mango of 500 gram? assume the seeds are proportional!!

AdnanOctober 24, 2015 at 6:38 amBrilliant article Kalid, it gives wings to ur imagination!!!! i would suggest adding some practical uses of this theorem as i had came to this page looking for some daily uses of the theorem but ended up getting amazed by it… :-D

ChuckApril 23, 2016 at 3:19 pmExcellent…very enjoyable trip

ChuckApril 23, 2016 at 3:27 pmCan also derive special relativity equations using Pythagoras eqn

AhmedApril 30, 2016 at 5:23 amOne of the best articles I read about Pythagorean theorem!

What I found a bit unclear is: “50 inputs = 40 inputs + 30 inputs”

Because these are not equal, but their combined squares are!

50 = sqrt(((40^2)+(30^2)))

How to master mathematics – MotivatemeFebruary 18, 2017 at 6:46 pm[…] Pythagorean Theorem is not just about triangles. It is about the relationship between similar shapes, the distance […]

Greek Mathematics – Pythagoras – Site TitleFebruary 23, 2017 at 11:30 am[…] Azad, K (2008). Surprising Uses of the Pythagorean Theorem. Retrieved from https://betterexplained.com/articles/surprising-uses-of-the-pythagorean-theorem/ […]

Reading List for the Week ending April 21, 2017 – Reading DietApril 16, 2017 at 11:21 am[…] Surprising Uses of the Pythagorean Theorem […]

Laura PageMay 5, 2017 at 1:21 pmI always knew Pythagoras was cool! Thank you so much for this website!!

CarlosMay 14, 2017 at 6:28 amThe Pythagorean Theorem is forever . . .

AnnonimasJuly 12, 2017 at 10:42 amThe Pythagorean theorem sounds really cool and amazing

Danilo MorãesAugust 2, 2017 at 8:44 amHey man, I’ve been following your website for quite some time now and I’ve learned a lot from you! But I think your “Intuitive Look at The Pythagorean Theorem” is not so intuitive and it actually sounds like a circular argument. Finding the area of a square from its diagonal only makes sense if we agree the Pythagorean theorem holds true: the diagonal (d) is the side of a square made from adding the square of both sides (s) of this original square. So because we know the Pythagorean theorem is true, we know the area of a square of side d is actually s^2 * 2, hence we only need to divide it by 2 to get the area of 1 square, therefore d^2/2 = s^2. I think youre implying the theorem to prove itself, not really an intuition IMO. If you have any comments about what I just said Id be glad to hear it. Thanks!!

kalidAugust 2, 2017 at 10:28 amHi Danilo, the relationship:

Area = factor * (line segment)^2

that holds for any shape comes from similarity, not the Pythagorean theorem. But, if you want to find the area of the square in terms of the diagonal with pure geometry:

Take 4 squares (with the diagonal, d) and stack them like this:

The area of the large inner square that’s formed is d^2. The entire of the full outer square is 2d^2. The area of one quadrant (that we wanted) is 1/4 of the outer square, or (1/4) * 2d^2 = 1/2 d^2.

Pythagorean Theorem DayAugust 15, 2017 at 7:00 am[…] with some triangle shaped food! Use the theorem to determine if two mediums or a large pizza will be bigger, then revel in the delicious beauty of a triangle in every […]

GregSeptember 22, 2017 at 10:22 pmHi Kalid,

First, this is probably one of the single most intuitive explanations I have ever seen. Thank you.

However, something is still bothering me. I can feel the answer is close but I’m unable to articulate it and was hoping you could help.

Pythagorean theorem ultimately works because all angles in a triangle must add up to 180 degrees. If 1 angle is 90 degrees that guarantee the other two angles Angle A + Angle B = 90.

Angle A / 90 should be the percentage of how much Angle A is contributing to the required 90 degrees. My gut tells me that this value should be of some significance and should be able to be multiple or divide it with the hypotenuse or one of the sides to get a proportional value ( a value that in some way relates to the other sides)

However, it doesn’t work. In fact, it appears as though that any relationship between the angles and the sides of the triangle must contain a squared value. I have found no examples of a trigonometric identity that doesn’t square at least one value. Why is this? If they values squared have a proportional relationship then why don’t the values have a similar relationship prior to being squared?

kalidSeptember 29, 2017 at 12:24 pmHi Greg,

Thanks for the comment, great question. So, the standard Pythagorean Theorem requires angles of 180 degrees a plane (not sphere or other shape), but there is a different version if you’re on a sphere:

https://www.math.hmc.edu/funfacts/ffiles/20006.2.shtml

So, the nature of the relationship does depend on the actual “geometry of the universe”.

For the squared vs. direct relationship, my intuition is that to compare things moving in different dimensions, you need some notion of “self-similarity”. I can’t really directly compare the GDP of Japan (Yen) and the USA (USD), but I can compare their relative rates of growth. By taking the square of a number we are sort of getting a notion of the size of a self-similarity, which can be compared (how similar is A to itself, vs. how similar is B to itself?). I think this would be a good follow up article!

daily 12/15/2017 | Cshonea's BlogDecember 15, 2017 at 12:31 pm[…] Surprising Uses of the Pythagorean Theorem – BetterExplained […]

Mike LeeJuly 7, 2018 at 1:00 amThis article has enlightened my understanding that the Pythagorean theorem is just addition played out in two dimensions. The notion of + requires that both operands are similar, one cannot add apples and oranges, it doesn’t make sense. Adding similar shapes is what the theorem allows us to do, thereby enabling the decomposing of area into simple numbers.