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Pythagorean Theorem

Updated on September 6, 2016

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What was Pythagoras thinking?

One of the rules you learn in Geometry is the Pythagorean theorem. This states that the sum of the square of the two legs of a right triangle is equal to the square of the hypotenuse.

What exactly does that mean?

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What is the Pythagorean Theorem?

Looking at the diagram on the right, you can see a right triangle. The Pythagorean theorem states that if you add the squares of the two legs equal the square of the hypotenuse (the side of the right triangle opposite the right angle -- this would necessarily be the largest side of the triangle).

This means that if you have the lengths of any two sides, you can find the third. For example, you have a right triangle and you know that one legs is 5 and the hypotenuse is 13. You take the formula and substitute in --

step 1 -- 5 squared + B squared = 13 squared

step 2 -- 25 + B squared = 169 --

step 3 -- B squared = 169 - 25

step 4 -- B squared = 144

step 5 -- B= the square root of 144 or 12

The Pythagorean Theorem: A 4,000-Year History
The Pythagorean Theorem: A 4,000-Year History
By any measure, the Pythagorean theorem is the most famous statement in all of mathematics, one remembered from high school geometry class by even the most math-phobic students. Well over four hundred proofs are known to exist, including ones by a twelve-year-old Einstein, a young blind girl, Leonardo da Vinci, and a future president of the United States. Here--perhaps for the first time in English--is the full story of this famous theorem.Although attributed to Pythagoras, the theorem was known...

Pythagoras's Diagram

What this theorem represents

When Pythagoras developed his theorem, he was talking about the area of squares. The diagram at the right represents what Pythagoras meant. He meant that the area of a square with sides the length of one side of a triangle (side A, in magenta, length 3) added to the area of a square with sides the length of the second side of the triangle (side B, in blue, length 4) would add up to the area of a square with sides the length of the hypotenuse (side C, in purple, length 5).

What this means in our example, the area of the magenta square (with a side length of 3 and an area of 3X3 or 9) added to the area of the blue square (with a side length of 4 and an area of 4X4 or 16) equals the area of the hypotenuse (or purple square -- with a side length of 5 and an area of 5X5 or 25). 9+16=25.

This can help you find the length of the third side of a right triangle when you have the other two.

Compugraph Designs Arts Now Site

"Arts Now" is another "Print on Demand" site. They have a nice collection of clocks and watches, including the one pictured here (with math symbols on it). Click on the picture to see the entire math clock collection on the Arts Now site.

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Pythagorean Triangles (Dover Books on Mathematics)
Pythagorean Triangles (Dover Books on Mathematics)
The study of the arithmetical properties of triangles dates back to ancient Greece, and possibly beyond. This classic text, written by a distinguished mathematician and teacher, focuses on a fundamental cornerstone of elementary geometry, the theorem of Pythagoras, and its applications. Unabridged republication of the edition published by the Graduate School of Science, Yeshiva University, New York, 1962. Translated by Dr. Ambikeshwar Sharma.
Math Travel Mug
Math Travel Mug

Compugraph Designs' Printfection Store

In addition to our Cafe Press and Zazzle sites (see modules above), we also have a store on "Printfection" which includes cutting boards (good wedding or housewarming gifts), mugs and cups, tees, etc.

This travel mug only one of several math themed items at our store:

Compugraphd Printfection site

(Click on the picture to go directly to this product's page)

Compugraph Designs' Site on Printpop

I just discovered Printpop -- check out my entire portfolio or click on the graphic to see just this product (called "Black Math"). Check back periodically as new designs are uploaded.

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    • Blackspaniel1 profile image


      8 years ago

      Now go for three dimensions. Yes, it is exactly what you might think.

    • kpp2385 profile image


      8 years ago

      this is excellent information.. and lucidly explained

    • profile image


      8 years ago

      great info source

    • profile image


      8 years ago

      great info source

    • hlkljgk profile image


      9 years ago from Western Mass

      nice resource -thanks.

    • profile image

      Kenken99 LM 

      9 years ago

      Nice lens. Can you add in more information about using the Theorem to find unknown sides?

    • profile image


      9 years ago

      thx for the advice it's already implemented , if you need some advice in regards to marketing I can help. send me a note with your direct email so i won't have to reply on your lenses all the time. cheers :)

    • profile image


      9 years ago

      Great content I'll be reading some of your other lenses soon

    • LabKittyDesign profile image


      10 years ago

      Pythagoras also didn't believe in irrational numbers, yes? Didn't he throw one of his students overboard who suggested that the square root of two could not be rational when they were out sailing on the Aegean? Maybe our prof just made that up. Anyway, thumbs up from fellow Squidoo and Zazzle nerdz.

    • profile image


      10 years ago

      This is SO IN DEPTH!! Great coverage of Pythagorean Theorem, a very important and oft-used concept in math. I had one thing to add as I just wrote a blog post highlighting a video we have instructing kids on the Pythagorean Theorem. I think it's a great explanation and is a good companion to all the links and information you have spent so much time gathering!!

    • joanhall profile image

      Joan Hall 

      11 years ago from Los Angeles

      I really like your math lenses. I'm adding them to a lensography I'm making for a homeschooling lens.


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