# How To Measure Any Distance With The Pythagorean Theorem

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.

It’s not about distance in the sense of walking diagonally across a room. It’s about any distance, like the “distance” between our movie preferences or colors.

If it can be measured, it can be compared with the Pythagorean Theorem. Let’s see why.

## Understanding The Theorem

We agree the theorem works. In any right triangle:

If a=3 and b=4, then c=5. Easy, right?

Well, a key observation is that a and b are at right angles (notice the little red box). Movement in one direction has no impact on the other.

It’s a bit like North/South vs. East/West. Moving North does not change your East/West direction, and vice-versa — the directions are independent (the geek term is orthogonal).

The Pythagorean Theorem lets you use find the shortest path distance between orthogonal directions. So it’s not really about right triangles — it’s about comparing “things” moving at right angles.

You: If I walk 3 blocks East and 4 blocks North, how far am I from my starting point?

Me: 5 blocks, as the crow flies. Be sure to bring adequate provisions for your journey.

You: Uh, ok.

## So what is “c”?

Well, we could think of c as just a number, but that keeps us in boring triangle-land. I like to think of c as a combination of a and b.

But it’s not a simple combination like addition — after all, c doesn’t equal a + b. It’s more a combination of components — the Pythagorean theorem lets us combine orthogonal components in a manner similar to addition. And there’s the magic.

In our example, C is 5 blocks of “distance”. But it’s more than that: it contains a combination of 3 blocks East and 4 blocks North. Moving along C means you go East and North at the same time. Neat way to think about it, eh?

## Chaining the Theorem

Let’s get crazy and chain the theorem together. Take a look at this:

Cool, eh? We draw another triangle in red, using c as one of the sides. Since c and d are at right angles (orthogonal!), we get the Pythagorean relation: c2 + d2 = e2.

And when we replace c2 with a2 + b2 we get:

$\displaystyle{a^2 + b^2 + d^2 = e^2}$

And that’s something: We’ve written e in terms of 3 orthogonal components (a, b and d). Starting to see a pattern?

## Put on your 3D Goggles

Think two triangles are strange? Try pulling one out of the paper. Instead of lining the triangles flat, tilt the red one up:

It’s the same triangle, just facing a different way. But now we’re in 3d! If we call the sides x, y and z instead of a, b and d we get:

$\displaystyle{x^2 + y^2 + z^2 = distance^2}$

Very nice. In math we typically measure the x-coordinate [left/right distance], the y-coordinate [front-back distance], and the z-coordinate [up/down distance]. And now we can find the 3-d distance to a point given its coordinates!

## Use Any Number of Dimensions

As you can guess, the Pythagorean Theorem generalizes to any number of dimensions. That is, you can chain a bunch of triangles together and tally up the “outside” sections:

You can imagine that each triangle is in its own dimension. If segments are at right angles, the theorem holds and the math works out.

## How Distance Is Computed

The Pythagorean Theorem is the basis for computing distance between two points. Consider two triangles:

• Triangle with sides (4,3) [blue]
• Triangle with sides (8,5) [pink]

What’s the distance from the tip of the blue triangle [at coordinates (4,3)] tot the tip of the red triangle [at coordinates (8,5)]? Well, we can create a virtual triangle between the endpoints by subtracting corresponding sides. The hypotenuse of the virtual triangle is the distance between points:

• Distance: (8-4,5-3) = (4,2) = sqrt(20) = 4.47

Cool, eh? In 3D, we can find the distance between points (x1,y1,z1) and (x2,y2,z2) using the same approach:

$\displaystyle{distance^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

And it doesn’t matter if one side is bigger than the other, since the difference is squared and will be positive (another great side-effect of the theorem).

## How to Use Any Distance

The theorem isn’t limited to our narrow, spatial definition of distance. It can apply to any orthogonal dimensions: space, time, movie tastes, colors, temperatures. In fact, it can apply to any set of numbers (a,b,c,d,e). Let’s take a look.

## Measuring User Preferences

Let’s say you do a survey to find movie preferences:

1. How did you like Rambo? (1-10)
2. How did you like Bambi? (1-10)
3. How did you like Seinfeld? (1-10)

How do we compare people’s ratings? Find similar preferences? Pythagoras to the rescue!

If we represent ratings as a “point” (Rambo, Bambi, Seinfeld) we can represent our survey responses like this:

• Tough Guy: (10, 1, 3)
• Average Joe: (5, 5, 5)
• Sensitive Guy: (1, 10, 7)

And using the theorem, we can see how “different” people are:

• Tough Guy to Average Joe: (10 – 5, 1 – 5, 3 – 5) = (5, -4, -2) = √(25 + 16 + 4) = 6.7
• Tough Guy to Sensitive Guy: (10 – 1, 1 – 10, 3 – 7) = (9, -9, -4) = √(81 + 81 + 16) = 13.34

We can compute the results using a2 + b2 + c2 = distance2 version of the theorem. As we suspected, there’s a large gap between the Tough and Sensitive Guy, with Average Joe in the middle. The theorem helps us quantify this distance and do interesting things like cluster similar results.

This technique can be used to rate Netflix movie preferences and other types of collaborative filtering where you attempt to make predictions based on preferences (i.e. Amazon recommendations). In geek speak, we represented preferences as a vector, and use the theorem to find the distance between them (and group similar items, perhaps).

## Finding Color Distance

Measuring “distance” between colors is another useful application. Colors are represented as red/green/blue (RGB) values from 0(min) to 255 (max). For example

• Black: (0, 0, 0) — no colors
• White: (255, 255, 255) — maximum of each color
• Red: (255, 0, 0) — pure red, no other colors

We can map out all colors in a “color space”, like so:

We can get distance between colors the usual way: get the distance from our (red, green, blue) value to black (0,0,0) [formally labeled delta e]. It appears humans can’t tell the difference between colors only 4 units apart; heck, even 30 units looks pretty close to me:

How similar do these look to you? The color distance gives us a quantifiable way to measure the distance between colors (try for yourself). You can even unscramble certain blurred images by cleverly applying color distance.

## The Point: You can measure anything

If you can represent a set of characteristics with numbers, you can compare them with the theorem:

• Temperatures during the week: (Mon, Tues, Wed, Thurs, Fri). Compare successive weeks to see how “different” they are (find the difference between 5-dimensional vectors).
• Number of customers coming into a store hour-by-hour, day-by-day, or week-by-week
• SpaceTime distance: (latitude, longitude, altitude, date). Useful if you’re making a time machine (or a video game that uses one)!
• Differences between people: (Height, Weight, Age)
• Differences between companies: (Revenue, Profit, Market Cap)

You can tweak the distance by weighing traits differently (i.e., multiplying the age difference by a certain factor). But the core idea is so important I’ll repeat it again: if you can quantify it, you can compare it using the the Pythagorean Theorem.

Your x, y and z axes can represent any quantity. And you aren’t limited to 3 dimensions. Sure, mathematicians would love to tell you about the other ways to measure distance (aka metric space), but the Pythagorean Theorem is the most famous and a great starting point.

## So, What Just Happened Here?

There’s so much to learn when revisiting concepts we were “taught”. Math is beautiful, but the elegance is usually buried under mechanical proofs and a wall of equations. We don’t need more proofs; we need interesting, intuitive results.

For example, the Pythagorean Theorem:

• Works for any shape, not just triangles (like circles)
• Works for any equation with squares (like 1/2 m v2)
• Generalizes to any number of dimensions (a2 + b2 + c2 + …)
• Measures any type of distance (i.e. between colors or movie preferences)

Not too bad for a 2000-year old result, right? This is quite a brainful, so I’ll finish here for today (the previous article has more uses). Happy math.

## Questions & Contributions

1. Bob says:

Very, very cool post.

I manage a data warehouse and get called in to analyze product data periodically — I see where I might be able to use this to some great effect.

I really need to go back to school and get a degree in statistics — if only school paid as well as data warehousing!

2. Kalid says:

Thanks Bob, glad you liked it. Yes, there are tons of applications — if you have a warehouse full of product data, I’m sure there’s some interesting trends/groupings you’ll be able to pull up

3. This post pointed me to a possible issue in the new version of my political simulation game I’m currently developing: I use ideological axes to rate political parties’ positions, and I currently determine their similarity by taking the average difference for each axis. Of course, I need to average the squares of the differences!

4. Jesper says:

Wow, this is very cool. I remember being taught this theorem in 4th grade math, but I never revisited it.

Now that I’m working on some things that could really benefit from collaborative filtering, I’m very happy to have come across this post!

5. oele says:

This is just one of any number of distance functions. You might want to look up the others :

http://en.wikipedia.org/wiki/Metric_(mathematics)

Especially manhattan distance is useful. But it is by no means the only one. You describe euclidean distance, you also have manhattan distance, hamming distance, jaccard distance, accoustic metrics, …

6. Josef Svenningsson says:

Just as oele wrote there are more types of metrics. I’m not saying you should include any of that in this article but some of the things you say are simply not true, due to the fact that there are other metrics. Here’s where it goes wrong :”Measures any type of distance”. You should change this statement or remove it.

Apart from that, a very nice article!

7. Ddd says:

I’m shocked that so many people didn’t know this. This is truly elementary material, and if articles like this need to be written for coders, the education system must be in a very sorry state.

8. Andrew White says:

Excellent, I knew all about Pythagoras theorem but I never thought about repeated applications and alternative uses.

Thanks, Andy.

9. @Wouter, Jesper: Thanks, I hope you have luck in applying it to your area

@oele: Thanks for the info — that may be a subject of a follow-up article.

@Josef: Thanks for the tip. I can rephrase to mean “provides one way to measure distance”. The key point is that the Pythagorean Theorem can generalize to N dimensions (it isn’t the only such theorem that does so).

@Ddd: Everyone has to start somewhere. And there’s always ways to look at “old” results in a new light.

10. Ddd said, “I’m shocked that so many people didn’t know this. This is truly elementary material, and if articles like this need to be written for coders, the education system must be in a very sorry state.”

To start a dialog, I think you are confusion a computer science education with that of what coders do now a days. There are many programmers who have not had formal mathematical or computer science education. Therefore, there will be some that don’t have this “basic” skills as you call it.

So the computer science education (at least in some schools) has nothing to do with some people not knowing basic vector algebra. In other words, coders come in different flavours.

11. Kalid says:

Thanks Jose, my thoughts exactly. Everyone starts in a different place: some people are designers who do programming on the side, others are programmers who do design on the side.

For some reason, math topics seem to encourage people to show off how much they know (and this attitude is partially why math isn’t a well-loved subject).

Me, I just like to write about what I find interesting, hoping other people enjoy it too.

12. Matt says:

Great post. I have used the multi-dimensional distance calculation to estimate similarity between all sorts of things, using different measures. The only thing to be careful about is scaling. The distances (dis-similarities) need to be re-weighted (or normalized), if scales are off.

It yields really interesting results.

13. Kalid says:

@Matt: Thanks, glad you liked the post. You’re right, the multi-dimensional distance is a good starting point but needs to be tweaked/scaled appropriately.

@Karl: Thanks for the tip — there’s so many interesting uses.

14. I was in a math class a few years ago where we proved the Pythagorean theorem multiple times in multiple ways. It was incredible to see how it works and why it works. I love the fact that math can be both incredibly complex and extremely simple all at the same moment.

15. Gav says:

I literally stumbled onto this post using StumbleUpon and think this is very cool. Like Bob I also work in Datawarehousing and on reading this post realised this concept could easily be applied to RFM (Recency, Frequency, Monetary) measures to identify groups for targeted marketing campaigns. Thanks.

16. Sylvain Romiguier says:

Great article ! Thanks to share your knowledge.

17. Dhwani says:

Hi People,

Please correct me if I am missing something here.

As stated above this theorem can apply to any orthogonal dimensions. In that sense if we need to measure the distance between people’s preference about movies then we have to first ensure that the three questions
1. How did you like Rambo? (1-10)
2. How did you like Bambi? (1-10)
3. How did you like Seinfeld? (1-10)
are actually orthogonal. That is to say, there is no relation between the response that any random person would give for any two of the three questions. For instance if any one who like Bambi is also likely to like Seinfeld, then these questions are not orthogonal. I am not a maths/stats expert but I feel this can be checked by calculating the correlation coefficient between the responses to any two questions. The coefficient for all pairs should be close to 0, only then we can apply Pythagoras Theorem.

On similar lines we can say that temperatures on Mon through Friday are not likely to be orthogonal. Rather they are likely to be linear. If it has been hot on Monday, Tuesday is more likely to be hot than cold. The same is the case with Age, Height and Weight.

18. Kalid says:

@Min: I agree, it can be really eye-opening to see old results in a new light, and how “simple” equations can be responsible for so much.

@Gav: Thanks for dropping by, I hope you find it helpful.

@Sylvain: You’re welcome, it’s a lot of fun and I end up learning so much when revisiting topics I thought I “knew”.

@Dhwani: Great point. Strictly speaking, the distance function works best when the quantities are orthogonal (i.e. movement in one direction has no impact on the other), but “correlation” shouldn’t be a factor. (I’m not a stats expert either, so don’t quote me on this ).

For example, it may be that everyone who moves North also moves East for some reason or another. But it doesn’t change how *far* they are from the starting point.

Similarly, there may be a correlation between liking Rambo and liking Bambi, but it doesn’t change the absolute distance between the preferences.

Having a correlation may cause preferences to be “clumped” along patterns (like being in the North-East) vs. being randomly distributed, but I think this is a separate issue from finding distance.

19. Thomas says:

Dear Kalid,

You write:
“For example, it may be that everyone who moves North also moves East for some reason or another. But it doesn’t change how *far* they are from the starting point.”

No, it doesen’t. But it changes how you can compute that distance. As an example: if you move first 3 units east and then 4 north, you are 5 units from the origin, according to Pythagoras. But if you first move 3 units east, and then 4 units north/east according to your comment, you are not 5 units from the origin. (you are slightly longer from the origin than 5 units in that case).

20. Thomas says:

I forgot to say that I agree with Dhwani (post 18): it has everything to do with correlation.
Therefore, the description in you post is not correct for correlated data.

I also agree with oele and Josef (post 5 and 6) – there are hundreds of distance measures out there, each useful in a differenet application. While Euclidean distance is convenient and easy to compute, it may in many cases give a picture that is far from correct.

And Wouter Lievens (post 3): be careful, the measure you already use may or may not be better than Euclidean distance. There is nothing that says that Euclidean distance is the best measure for such cases.

21. Kalid says:

@Thomas: Thanks for dropping by. I should have been more clear by what I meant with the term correlation.

If there’s a “physical” reason why moving East would also move North, then yes, distance can’t be measured using the theorem.

If moving East pushed you North as well (due to a strange gust of wind) then you couldn’t measure distance accurately, as someone moving East would have “free” distance in the North direction as well.

If there’s simply a “psychological” reason (i.e., everyone decides to move along a line where x = y) then the theorem can work. Imagine people all decide to walk along a path directly North-East; they aren’t *required* to, but just happen to do so. In that case there is a 1-1 correlation between their North and East location, but distance can still be computed fine.

@Clinton: Not sure what you mean about source code!

22. I don’t think kids realise when they learn this how important this is – I just used Pythagoras last week to calculate the area in my bay window. My girlfriend thought I was a genius.

23. DJ says:

For LLLL,

What was the shape of your bay window?

24. This is a but a special case. The special portion is the fact that space must be euclidean, that is flat. This will fail if used in a curved space (like the one we live in) over a long distance. What you really want to talk about is the Metric Tensor on a Riemannian manifold.

25. In 3 dimensions the familiar pythagorean theorem is a special case of the law of cosines (http://en.wikipedia.org/wiki/Law_of_cosines) which contains the inner product. This formula does NOT extend to higher dimensions in a trivial way (the inner product term gets more complicated). As always be careful in generalizing a special case!

26. Kalid says:

@Chris: Thanks, glad you liked it.

@George: Thanks for the info, it may be helpful for people who want to dive into more details [I may do a separate follow-up on these topics].

27. Martín says:

Good job. Just add a warning against abuse.

“Qualitative is nothing but poor quantitative.”
Rutherford

“Not everything that counts can be counted, and not everything that can be counted counts.”
Einstein

My heart is with old man Bert here. I’m sick of meaningless measures overdose. Think a moment before you produce your whatevermeter.

28. Hi Martin, thanks for dropping by. Yes, when you have a hammer everything starts looking like a nail.

It’s good to bend your brain with new uses, but all things in moderation. Don’t go trying to rank your favorite kid this way :).

Prior to reading your post, I had to determine if my 16 foot canoe would fit in the cheapest rental space. The dealer said (remarkably) that it might, if I could figure the length, width, height dimensions properly (I was surprised that he knew it could be calculated, and if so, why he was holding a job renting storage spaces). The result of my calculation (Pythagoras Theorem in 3 dimensions) predicted that I wouldn’t be able to close and lock the overhead door if the canoe were laid flat on the floor, but I might be able to it one end of the canoe were raised up to the ceiling in the storage space. Interesting post, and it certainly promises to go beyond the usual conceptual bounds of three dimensions.

30. Hi, I’m glad it was useful for you! (This is about as real-world as it gets).

Just be sure to leave a bit of buffer — the Pythagorean Theorem works well for things like strings or rods of zero-thickness; make sure the sides of the canoe don’t hit the corners

31. Anonymous says:

hello, very nice article. i really enjoyed it. my friends are appreciating too [:)]

32. Anonymous says:

33. Kalid says:

34. al says:

I just discovered this site and love it! I am studying engineering in school, but I hate to learn things (especially math) without really understanding them- this page has really opened up a new way for me to think of dot products in 3dims, as well as other concepts I have learned from a different angle. thanks!!!

35. Kalid says:

Hi Al, thanks for the comment! Yes, I also hate learning things without a deep understanding — I’m glad you’ve found the articles useful

36. sridhar chitta says:

Could someone let me know how to measure the distance of a function like y(t) = at^2 + bt + c where t is time and a, b and c are constants ?

37. Kalid says:

Hi sridhar, I’m not quite sure of the question. If distance is one-dimensional and based on time, then you can simply take the difference [y(t2) – y(t1)].

38. jess. says:

i have maths homework –
it sais
find the distance between these two poiints by using the pythagorean theorem
and i still have no idea how do to it

39. Kalid says:

Hi Jess, for problems like this you need to plug in values for a and b, and use the equation

c = square root of(a^2 + b^2). You might want to ask your math teacher to show with a diagram. Hope this helps.

Wow that is intense

41. michael says:

How do you find an angle using triginometry?

42. Kalid says:

@jade: Thanks for the comment, some of this stuff can be brain-bending at first.

@michael: You can find the angle in a right triangle using the arc tangent (called atan). On a calculator you can do atan(b/a) where b is the height and a is the width. In a 3-4-5 right triangle (a=3, b=4, c=5), you could do atan(4/3) and get 53.13 degrees.

43. Looks like a math trick Wonder if this very technic is utilized by some applications, like Photoshop.

44. Hi collector, yep, I’m sure Photoshop must use color distance in some of its transformations (brightness/contrast/color shifting).

45. Great! Was led to this post while searching for better (graphical) explanations of measuring distances. Thanks!

46. Thanks Hrishikesh, glad you enjoyed it.

47. Lowell says:

The color example was exactly what I was looking for. I have been trying to explain this to a sales person for months. (I work for a machine vision company.) Your explanation is perfect. Thank you!

48. Kalid says:

@Lowell: I’m glad you enjoyed it! Happy you were able to make use of it — I like the color example because it appears to have “nothing” to do with geometry.

@Grace: You’re welcome.

49. Stephen Nagy says:

Here’s my question: I’ve got a triangle who’s sides are 1 (one), a and h.

h is orthogonal to a’s plane.
The side called ‘1’ is always 1 long.
a varies from 1.

I know 1 and a and need to solve h.

It’s OK if h is either positive or imaginary. I’ve even thought of putting a and 1 onto quat type i,j,k.

Am I overlooking something obvious?

Steve.

50. Stephen Nagy says:

I’ve got a question that this post is close to:

I’ve got a triangle side’s lengths are 1 (one), a, and h.
h is orthogonal to a
a varies 1
i need to solve h.

it’s OK if h is real or imaginary or I could even use quaternian space for the whole thing. Any hints?

(sorry if this is a duplicate, but scripts were disabled on my first post attempt.)

Steve.

51. Kalid says:

Hi Steve, thanks for the message. I’m not sure I understand the question — what does a varies 1 mean? As you change a, h must change to satisfy the relation a^2 + h^2 = 1.

Hope this helps.

52. Anonymous says:

thanks so much!

53. Nobody says:

I’m not sure about you, but I’d personally take that as a compliment.

Anyway, nice article! I do like the colors example.

54. Kalid says:

@Nobody: Thanks, glad you liked it!

55. edgar says:

this is very confusing for me
i am cunfuzzled
how do u find the distance of water rockets using
that method??????? ;/

56. MIGUEL says:

DIZ IS HARDCORE 😉 ;?

57. Dave says:

hence comes the phrase:
”i don’t know the answer, but i know its got something to do wiith triangles”
haha this is some clever stuff 😀

58. Smarti says:

I sew. Sometimes I have to make bias. It means cutting strips on the ‘c’ edge of the fabric. I want to know the formula I should use to calculate how much 1.5in strips I can get this way out of a specified size square of fabric. I have to cut the square in two diagonally so it makes two right angled triangles. But I don’t know how to calculate the surface so I can figure out a total length of 1.5in strip if they were all put end to end. Can you help? Use a 54in square as an example. But it could be 45 or 60 depending on the width of the fabric purchased.

59. peter says:

Was the correlation issue answered regarding the netflix example? I assume this is the orthorganal part of the problem, but I am not sure I understand how to determine orthorganality in this case. Likewise, is it accurate to say the PT is about “any” distance? The title of the post says “measuring any distance” but seems more like “any orthoganal vectors” would be the accurate phrase, no?

60. ferret says:

i know everything you said now on the posts! your math is so easy to learn and complete thanks

61. Kalid says:

62. dumbo says:

63. Kalid says:

64. sajid says:

i want to ask a question.
when a man watch a thing from distance how can he measure the thing size. please send the answer to my ID.

65. Ryane says:

how do you use the pythagorean theorem to express the distance as a function of time. Here’s the problem: A ladder 25 ft long leans against a vertical wall with its foot on level ground 7 ft from the base of the wall. If the foot is pulled away from the wall at the rate 2 ft/s, express the distance (y ft) of the top of the ladder above the ground as a function of the time t seconds in moving.

66. Taylor says:

i believe the Pythagorean theorem shows up readily in sum of squares calculations, like for instance with ANOVA. This allows statisticians to determine whether there is significant variability between any pair of treatments.

67. Kalid says:

@Riscelle: You’re welcome!

68. rahwa says:

can you make one quation?

69. Joy says:

Thx khalid for such as awesome insight into vectors, it helped me really become friends with mechanics 😛 the intuition doesn’t really help in solving questions, but it gives a lot of confidence when solving any questions regarding vectors, your insight helped me understand why V(final)=V(initial) + at which is not a really big deal but the formula really becomes quite freaking obvious.

70. Kalid says:

@Joy: Awesome — one of the best things in math is when ideas that were once complicated become “obvious” :).

71. Note that as the number of dimensions get higher than 1, there is more than one way to describe the distance between two points.

A common one is the Euclidean distance defined by d² = (x2 – x1)² + (y2 – y1)² (cf the article). But you can also define d’ = abs(x2 – x1) + abs (y2 – y1) or d” = max(abs(x2-x1), abs(y2-y1)) (abs(x) is the absolute value of x, ie the value without the sign).

d’ is quite important in cities such as New York where all streets form a grid : the distance from one point to another is the number of blocks in a direction (ie abs(x2-x1)) plus the number of blocks in the other direction (ie abs(y2-y1)).

72. Anonymous says:

I understand how the pythagorean works and what it does, but I don’t understand WHY. I’ve been searching for an intuitive explanation for why orthagonal components are related in squared numbers. It may have to do with the fact that they one dimensional figures that are related in the second dimension

73. Brianna says:

My teacher didnt have time to explain this to me.. I haven’t really been in class lately.. And I’m stuck .. i feel like i dont get Pythagorean triangles… :/

74. Rishabh says:

Hi i want to how we can make a square that has same diagonal also for eg. if in a square all sides r equally then the diagonally must also be the same as square side pls, tell

75. toyin says:

how can i use concept mapping to teach rightangle triangle showing the chart.

76. Anonymous says:

hi i still dont get it

77. there is a simple to check distance by triangle. Only needs pencil, ruller, and paper. No need protractor nor compass. No need to get close to object. Only one distance must be known for scale, the rest can be calculated. This method is using 2 similar triangle, both have same angles but one triangle is smaller (drawing), the other is bigger (imaginery in landscape). Ratio of sides of both triangles is equal to drawing scale. Thus distance calculation can be made by measuring triangle drawing then multiplied by scale.
This method can be used to create scale map, and to calculate height of an object from a distance. Measurement is accurate enough to check landscape distance, calculate landscape area, landscape planning, etc. It is so simple that you can play treasure hunt game by map produced.

http://maruzar.blogspot.com/2011/12/measure-height-from-distant-with.html

78. robert hahn says:

Fascinating presentation on the implications of the pythagorean theorem. Perhaps you can answer MY question? In the 6th century BCE, the time of Thales and Pythagoras, the case can be made that these thinkers were searching for a unity that underlies all things, and here, the geometrical figure that underlies all other geometrical figure. The pythagorean theorem might be seen to reveal the right triangle as the basic unity. I’m stuck, however, in constructing an argument that: All (Euclidean) space is reducible or expressible as rectilinear figures (and hence the unity that underlies is the triangle of which all rectilinear figures can be reduced to). Can you produce THAT argument?

79. sarah says:

it is very useful & awesome

80. Oliver Marsh says:

Great Post!!!

I think you can apply the same idea to how standard deviation is computed.

81. AsVHEn says:

Nice post, when you have enough material you must do a book.
I have only one tip, you should use an 0 based scale (instead of 1 in “1. How did you like Rambo? (1-10)”) to rate things. 1-10 can lead to misunderstanding.

82. Jeremy Weiss says:

Another disappointing article. This is science not intution. Intuition is about feeling.
Physicist David Bohm suspected a deep relationship between the Pythagorean theorem and Green’s Theorem. He had great intuition and a deep feeling for this. But neither intuition nor feeling are found in this article. The meaning of orthogonal components is, sadly, never expressed. In the end I felt nothing.

83. Stephen Ramos says:

I finally noticed why the standard deviation formula works, it’s just the distance formula for multiple points.
First you find the average or the mean of all the points, then we figure out what the difference is from each point to the mean, and finally find out what the average distance is between each variation.

So to break it down were finding how much does each point vary from the mean so if we had two points a and b to find how much they vary we will use (a-mean) and (b-mean) then we find the how much does each point vary on average (a-mean)+(b-mean)/n where n is the amount of points and finally we find the distance or the amount that each point varies with each other on average. sqrt((a-mean)^2+(b-mean)^2/n).

84. kalid says:

@Stephen: Exactly! It’s the average distance from the center of the population.

85. Bob Roberts says:

Hi Kalid! How would you use the Pythagorean Theorem to find the distance travelled between Angola Africa to Newport News, Virginia? Thanks a bunch!

86. kalid says:

Hi Bob, great question. Unfortunately, the Pythagorean theorem only works on a plane, not on a sphere or globe. In that case, there are other formulas to find the distance on a sphere (you have to take the curvature into account).

It’s called finding the “Great Circle” distance, and for that, you use the longitude and latitude coordinates. Hope this helps!

87. Bg says:

This is so interesting. Thank you for sharing.

88. Hans says:

Kalid,

Could you show me the maths for this line:

Tough Guy to Average Joe: (10 – 5, 1 – 5, 3 – 5) = (5, -4, -2) = 6.7

How do you get to (5, -5, -2) = 6.7?

Thanjs

Hans

89. Hans says:

Actually, by (surprise/surprise) reading the artice, I see how you did it.

$\displaystyle{a^2+b^2+d^2=e^2}$

Thanks

Hans

90. kalid says:

Hi Hans, glad you worked it out! I’ll update the article to make that step more clear.

91. Sairam S says:

I have never had a background in maths, but am right now trying to learn statistics (because it would help my career). The one place where I have learnt a lot is this website. I totally believe you must be regarded a mathematical god for laymen like me in the current times. Thank you very much. I have got so much help from here. Thanks a ton.

92. Sairam S says:

I tried to think hard but couldn’t get what you guys – Stephen Ramos & Kalid (Comment # 90 & 91) exactly meant when you relate standard deviation with the Pythagorean Theorem! So, should I be imagining the mean to be the the Origin/ Vertex of the different axes and the difference of each point from the mean to be the co-ordinates describing the one point hanging in the n-dimensional space? Does that make sense? I’m unable to proceed any further with this thought!

93. Ruchira says:

No one ever told Pythagorean theorem is not about measuring distances between two points. The theorem itself came into existence even before Pythagoras in India due to it’s application in construction of homes and other buildings. :)! You are assuming way too much. There is nothing about triangles here. What people got to do with triangles anyway, so the point was it was always there for it’s application in building constructions. Then your idea of measuring preferences sounds really meaningless. It may give an average graph in certain system but calling it distances seems really thoughtless! Sorry to say this!

94. Sairam S says:

@Ruchira – “Then your idea of measuring preferences sounds really meaningless. It may give an average graph in certain system but calling it distances seems really thoughtless!” have you heard of how ‘Cluster Analysis’ is done in Statistics. If you know that, you wouldn’t find it meaningless really. This would in fact make a lot of sense to you!

95. Interesting that trolling through “the common core” (not a dis nor a promote just a comment) that the sense that a vector can be anything beyond 2 (or perhaps) 3 dimensional is completely missing. Good thing the Kalid is not just a common thinker.

96. 1) An interesting exercise with Pythagoras. Here are a couple of items that you should enjoy, Kalid Azad, that I unravelled in 2000 having retired and been stimulated by a book on 3D geometry.
2) Take a perfect cube, slice off one corner and discover that the SUM of the squares of the AREAS of the three ORTHOGONAL TRIANGLES is equal to the square of the AREA of the SLICED TRIANGLE. I call that Pythagoras in 3D.
3) Create an irregular tetrahedron with 4 non-identical triangular surfaces. Take a cube and slice off 4 corners so that each SLICED TRIANGLE is an exact match to the TETRAHEDRON TRIANGLES and ‘stick’ them on those surfaces to make a stellated tetrahedron. Discover that the SUM of the squares of the VOLUMES of THE CUBICAL CORNERS is equal to the square of the VOLUME of the CORE TETRAHEDRON. I call that Pythagoras in 4D. I found Lagrange’s four-square theorem, also known as Bachet’s conjecture to be useful in selecting suitable dimensions for manufacturing a wire model.
4) Just as the CORNER in 2) can be unfolded to make a 2D plan of all 4 TRIANGLES so the stellated construction IS the unfolded version of the 4D entity into 3D space. It does not have any magical properties to my knowledge and I was too cowardly to try to prove Pythagoras in 5D. Kind regards.

97. Raman Kannan says:

This is an awesome and insightful post for all math students …
and, also to those who have proposed corrections — tone might have been more positive but I enjoyed their message.
Thank you
Raman

98. Joe (Giuseppe) says:

The so called “Soleto’s map” dates back to beginning of 5th cent.BC being the first known topographic map in western culture.
It shows the end of italian peninsula (Salento) and unusually the towns and their names are marked with dots.
Close to the depicted area, the towns of Taranto and Metaponto, where Pitagora spent his last 10 or 15 years, are located.
I suspect that both Pitagora’s Theorem and his Tetraktys Diagram have been employed for the survey and the final map.
Wait for your comment. Many thanks. May the 1st, 2015.