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Pythagorean Theorem: The Theory of Pythagoras and How it Works

Updated on May 11, 2011

In school we tend to think that the Pythagorean theorem is just another stupid geometry problem that teachers throw at us. It's yet another headache that math gives us (as if we needed any more hard school subjects). And it's useless! All it does it solve a triangle's sides. Big whoop. Now I can figure out the dimensions of my triangular-shaped dog, Tripoodle.

Oh, but the Pythagorean theorem has many more uses apart from solving the dimensions of geometrically awkward animals. Read on to find out.

What is it?

The Pythagorean theorem, or Pythagoras' theorem, is a relationship between the sides of a right angle (90 degree) triangle.

It's written as an equation:

a2 + b2 = c2

Where a, b and c are the sides of a triangle, c being the side opposite to the 90 degree angle. It is also the longest side in a right triangle, known as a hypotenuse. The other sides, "a" and "b", are known as the legs or the catheti.

Right Angle Triangle

The right angle is denoted by the square shape in the angle C.

Angles A and B are acute angles - their values are less than 90 degrees.

In contrast, obtuse angles are angles with values greater than 90 degrees. There are no obtuse angles in right angle triangles.

This is all very dry and technical, but bear with me! There's some fun stuff up ahead!

How can you prove it?

There are many ways to prove the Pythagorean theorem. The book Pythagorean Proposition, by Elisha Scott Loomis, has 367 different ways of proving it.

Here is a great Hub that has a very simple and creative solution for the theorem. Don't get overwhelmed by the diagrams! It's actually very easy to understand.

But how do you solve a Pythagorean problem?

As long as you have two sides of a triangle, you can solve the remaining one using the formula.

a2 + b2 = c2

The only tricks here are that you have to identify which side is "c" or the hypotenuse, what sides you are given, and which side you are looking for.

Example 1 - Solving for Hypotenuse

How about a few examples?

To the side is a right angle triangle with an unknown variable "x". Now how do we go about solving this problem?

Step 1: Find out what information you are given.

We are given the length of two sides of the triangle, 3 and 4. We also know that it's a right angle triangle because of its 90 degree angle (shown by the square drawing).

Step 2: Identify the hypotenuse.

In this case, the hypotenuse is "x"; we know this because "x" is the longest side of the triangle and that it's opposite of the right angle.

This means that in the formula a2 + b2 = c2, the "x" value is the "c" value. It's given a different letter, but it means the same thing. It's just a variable for the hypotenuse.

This also means that the other two values, 3 and 4, that were given to us are the legs "a" and "b" in the formula. It doesn't really matter which is which, it'll turn out the same.

Step 3: Solve the equation!

Since we're looking for the hypotenuse "c", which is "x" in the diagram, and we're given 3 and 4 as "a" and "b" in the formula, we have everything we need to solve!

So writing it out:

(3)2 + (4)2 = x2

Now we expand the brackets by squaring the numbers:

9 + 16 = x2

We simplify that into:

25 = x2

And finally, we take the square root of the result:

x = 5

It's as easy as that! The hypotenuse of this triangle is 5.

Example 2 - Hypotenuse Given

How about another example?

Ok, this seems pretty straightforward, but it looks like we're given different values. Let's go through the steps to figure it out!

Step 1: Find out what information you are given.

We are given the length of two sides of the triangle, 10 and 6. And again, we know it's a right angle triangle because of the "square" diagram (it means it's perpendicular i.e. that it makes a right angle).

Step 2: Identify the hypotenuse.

In this case, the hypotenuse is 10; we know this because 10 is the longest side of the triangle and that it's opposite of the right angle.

This means that in the formula a2 + b2 = c2, the value 10 is the "c" variable.

This also means that the other value we're given, 6, is either "a" or "b". Again, it doesn't matter which.

Now we know that we're solving for one of the missing legs of the triangle. This is also given the variable "x" in the diagram.

Step 3: Solve the equation!

Since we're looking for the leg "x" and we're given10 as "c" and 6 as "a" (or "b") in the formula, we have everything we need to solve!

Writing it out:

x2 + (6)2 = (10)2

Now we expand the brackets by squaring the numbers:

x2 + 36 = 100

Now we rearrange the equation:

x2 = 100 - 36

Simplify it:

x2 = 64

And finally, we take the square root of the result:

x = 8

There we go! The missing "leg" of this triangle has a value of 8, the other "leg" is equal to 6, and the hypotenuse is equal to 10.

Wait a minute! That sounds oddly similar...

Look at our previous example - it has side values of 3, 4 and 5. Our second example has values of 6, 8 and 10. Our second triangle is just like our first triangle, but with all of its dimensions doubled!

These two triangles are known as similar triangles - they have the same shape, and they can scale up! You can shrink or expand each side by the same amount, and it'll look the same.

Ok I know how to use it, but what are some of its applications?

You mean other than figuring out the length of your Tripoodle's hypotenuse?

There are quite a few uses for the Pythagorean theorem. Construction, architecture and engineering have some common uses from the formula. It also comes useful in some physics problems, like vectors in kinematics and force/motion problems.

It's also a stepping stone to some of the more difficult mathematic problems. On this principle, you build up or combine it with more techniques like trigonometry.

The Pythagorean theorem is actually a special case of the cosine law, which applies to all triangles. This will be covered in a separate article.

Now why don't you get some practice with the following questions?

Question 1

Question 2

Question 3


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    • profile image

      Chelsia 3 years ago

      A wonderful job. Super helpful inotomarifn.

    • mrpopo profile image

      mrpopo 5 years ago from Canada

      Glad to hear it shuan!

    • profile image

      shuan molete 5 years ago

      This was amazing its explained better here I even got full marks for my pythagoras assignment thanks this was amazing

    • profile image

      shuan molete 5 years ago

      This was amazing its explained better here I even got full marks for my pythagoras assignment thanks this was amazing

    • profile image

      Akande Emmanuel 6 years ago

      Hello everyone,

      I would be conducting a seminar in my University on How to apply Pythagoras theorem in our practical life, professions and business, please if you have any information that can help, you can contact or send it to

    • mrpopo profile image

      mrpopo 6 years ago from Canada

      Sof, my apologies for the very delayed response, but I'm very glad this Hub helped you out!

    • profile image

      sof 7 years ago

      Hi mrpopo, just wanted to say thanks for this Hub, I think it's great when people post articles like these - I have found it really useful as I needed to revise for a maths test on various topics and my mind just went blank when it came to Pythagoras Theorem! Thanks to your well explained Hub, I passed all three questions above after having to practically teach myself Pythagoras again! Thanks, and I look forward to using more of your Hubs to teach myself other math techniques that I have forgotten! :)

    • mrpopo profile image

      mrpopo 7 years ago from Canada

      Hi pork, thanks for the comment! I'm glad you found the Hub useful! And hehe, tripoodle is pretty funny :)

    • pork22 profile image

      pork22 7 years ago

      This hub is done very well. Even though I learned all this in high school, there is nothing wrong with reviewing the information. You also provide definitions and hint at more advanced topics. The Tripoodle is too cute.

    • mrpopo profile image

      mrpopo 7 years ago from Canada

      Hey Ashley! Glad the Hub helped! I was thinking of doing these for various types of subjects because I always forget some concepts or tricks, especially in mathematics. I think I'll brush up on a textbook and perhaps write a few more :)

      Best of luck in college, and thanks for the comment!

    • profile image

      Ashley 7 years ago

      I felt I needed to tell you after reading your comments that I myself learned this stuff in the 9th grade, however I am now in college and needed help re-jogging my memory as I now work on college math. Your page was so helpful, and really helped me re-teach myself this matter. I am thankful that there are people like you out there who makes these pages which can be so useful in life.

    • mrpopo profile image

      mrpopo 7 years ago from Canada

      Haha, I'm not a fan of geometry either. But I absolutely love chemistry in general, and quantum mechanics just brings a new dimension to the subject. It's one of those things I would love to write about in the future. In fact, I was helping a friend out with the basics in naming compounds and the rules you follow, and I suddenly felt a nostalgia in learning the basics and building up on them to the more advanced thoughts. So hey, hopefully in the near future expect a Hub on the subject!

      The Pythagorean theorem is basic, yet powerful... and you would not believe how many university students tend to forget rules like that or the quadratic equation. But, it happens, and it's become a common occurrence in these educational systems.

      Some articles that I thoroughly enjoyed on the subject:

      If you have the time, check them out. They're quite interesting :)

    • lxxy profile image

      lxxy 7 years ago from Beneath, Between, Beyond

      Blargh. Despite my ability to understand mathematical concepts and figure stuff out like how long it would take a train to get to x place at y speed...I hate math. And I hate geometry more.

      I do possess the mental facilities to visualize (obviously, being a writer) but I don't have the patience.

      But ohh, do I love quantum mechanics. And mechanics, in general. I used to watch a PBS series on it, when I was up late enough to happen by it. Fun stuff.

      --guess I do well if I see the applications for it.

      But the Pythagorean theorem is as basic as you get...ahh, enough of my ranting. ;)

    • mrpopo profile image

      mrpopo 7 years ago from Canada

      Thanks Origin for the thoughtful comment! And it's true, it's very easy to forget these things. Just browsing through these concepts in Wikipedia and my old textbooks makes me go "wait, did I learn this before?". It's something that you just have to use everyday in order to keep it fresh in your memory, at least until it becomes second nature to you.

    • Origin profile image

      Origin 7 years ago from Minneapolis

      Wow, great explanation! When I first saw your hub title I had flashbacks of math class. I would still consider this knowledge to be helpful for people over grade 9. Why? Because a lot of people who have been out of school for years and need to figure this out and just don't remember would find this useful. I think a lot of people even though they've had advanced math classes in school don't remember every thing that they were taught, even if it's easy and simple for some people out there.

    • mrpopo profile image

      mrpopo 7 years ago from Canada

      That's quite all right Ant. I tend to do the same.

      You're right, I would be hard-pressed to find anyone who doesn't know Pythagorean theorem after grade 9. But I do know people who've forgotten the quadratic formula or who have forgotten how to do trigonometry, so it wouldn't really surprise me to know people who forget these things. If anything I just hope this helps either as an introduction or a refresher of the theorem.

      Ah ok, I guess that would fit into the category of proper use of Google. I just imagined Googling the answer as opposed to Googling how to solve the answer, which admittedly I still tend to do.

      That book seems very useful, and coming from Euclid it must be a good read. I'll check it out.

      That sounds complicated, and I can't remember that by memory. Maybe I've done that sort of problem before, but it sounds like calculus and I'm sort of limited in that knowledge. I definitely need to brush up on that. But any tutorials I make won't be exceedingly complicated, I'll have a few on relative velocities and two-dimensional kinematics, maybe dynamics and translational equilibrium problems... but I don't think I'll get past anything of grade 12, or at the most first year university. My background in physics is not the strongest, as my program is centered around biology, but I think I can find at least a few useful problems to talk about.

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      Ant 7 years ago

      Only after I had posted my rant (yes, it was a rant) I noticed this is your first week. Sorry for being too harsh.

      I just don't get how ANYBODY beyond Grade 9 doesn't know Pythagorean theorem. It's something as fundamental as using a spoon to feed oneself.

      When I said "to use more Google" what I meant was to use Google to find the excellent tutorials on anything a person doesn't know.

      A good book on anything concerning Euclidean geometry is Euclid's book called "Elements" written 2,000+ years ago:

      I'm eager to see what you're about to write about kinematics and vectors (OK, I'll admit it -- I wasn't very good in solving partial differential equations of functions of more variables in POLAR coordinates).

    • mrpopo profile image

      mrpopo 7 years ago from Canada

      @ prasetio - thanks for your comment bud, and coming from a teacher all the better. I appreciate your support.

      @ Ant - thanks for your comment, but to be honest I don't know what to make of it. It sounded more like a rant incorrectly aimed at my Hub, but I'll try my best to clarify any misconceptions you may have.

      First, I agree with you for the most part. Less Facebook would be optimal, but not necessarily more Google. It tends to make us overly dependent on it, and its answers are not always consistent or correct. It also tends to make it easy to find answers without even trying to solve it on your own, creating bad habits. I should know, I've been there. What I think you meant to say was more proper use of Google, but then again if you use Facebook wisely it can be a learning tool as well.

      This article was being aimed at whoever needs Pythagorean theorem help. I learned it in grade 9, so I assumed grade 9 students would benefit from it, but I'm sure anyone can benefit from this regardless.

      I don't think it's beyond anyone to understand triangles, which is why I made this article, in case you have no idea what Pythagorean theorem is or if you just want to brush up on it. If this is too easy for you, then that's great. Others have difficulty with it, however, so don't put them down for it. Note that my article is not the only one on seemingly simple tasks - if you'll look at the side, you'll find an article on how to calculate percentages.

      I was also aiming this to be one of the many excellent tutorials online that you so profoundly support. Since Hubpages had almost no Hubs talking about the Pythagorean theorem (a simple search of "Pythagorean theorem" yields mainly irrelevant results) I figured I would write one.

      This is also only my 3rd Hub, and my first week on Hubpages. If it tends to be a Hub that's just regurgitating well-known knowledge, then you'll have to forgive me. Like I said, I'm new here, and still learning.

      The main reason I wrote this Hub, however, is because it is a stepping stone to other Hubs. For example, in a future Hub I will be able to talk about kinematics or vectors that apply the use of Pythagorean theorem. I can then reference to this page in case they do not know or understand the Pythagorean theorem behind it.

    • profile image

      Ant 7 years ago

      I can't believe somebody just wrote 4 screens about Pythagorean triangles! Whom is this being aimed at? Grade 5 pupils?

      I'm afraid that if one can't understand triangles then they should start taking more art classes because any technical subject is obviously not going to be their cup of tea.

      With so many excellent tutorials online (including videos) there's today really no excuse not to understand anything that's being taught up to Grade 12.

      Maybe less Facebook and more Google is what kids today need.

    • prasetio30 profile image

      prasetio30 7 years ago from malang-indonesia

      It was fun learning Pythagorean here. As a teacher I also give this method for my student. Thank you very much, my friend. I rate this and thumbs Up.

    • mrpopo profile image

      mrpopo 7 years ago from Canada

      Luke, you must be one of the lucky few. I have never come across a Tripoodle, I've only seen them on documentaries. They are almost as elusive and mythical as Bigfoot or the Loch Ness monster...

    • profile image

      LukeEllington  7 years ago

      Tripoodles are an endangered species..I have only seen one or two in my lifetime