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An introduction into Pythagoras’ Theorem

Pythagoras’ Theorem


Pythagoras was born at Samos, in Greece, and lived from 580 to 500 B.C. He was a Mathematician who became famous for discovering something unique about right-angled triangles.

Let’s have a look at what he discovered…

So what is Pythagoras’ Theorem?

Well, he said that…

“For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.”

Consider squares being drawn on each side of a right-angled triangle…

See diagram below.

The formula for finding the longest side on a right-angled triangle is –   +  b² = c²   

The longest side I called the ‘Hypotenuse’ and is always the side opposite the right-angle.


Below is an example.

Find the length of the hypotenuse


  1. Add 3² = 9 and 4² = 16
  1.  The sum of the two numbers is 25
  2.  Finally, square root  x = √25

                                          x =  5cm


You can also find one of the shorter sides of a right-angled triangle by rearranging the formula.

The formula to find the shorter side is –   = c²   –  b²

The shorter sides are always adjacent to the right angle.

Below is an example.

Find the length of the side marked x


  1. Subtract 3² = 9 from 5² = 25 
  2. The answer is  = 16
  3. Square root x = √16

                             x =  4 cm



Pythagoras’ theorem can only be used in right-angled triangles.


Pythagoras’ theorem can be used to solve certain real-life practical problems, when a problem involves two lengths.


Below is a real-life Pythagoras problem.

A plane leaves Manchester airport and heads due east. It files 160 km before turning due north. It then files a further 280 km and lands. What is the distance of the return flight if the plane flies straight back to Manchester airport?

First, sketch the situation.

Using Pythagoras’ theorem gives:

   = 160+ 280km

    = 25600 + 78400 km

    = 104000 km²

So x = √104000 = 322 km     


Pythagoras’ theorem is interesting in its own right but is very important in other areas.

Here’s some Pythagoras trivia….

Trivia 1:

Pythagoras’ theorem has appeared in an episode of The Simpsons, when it  was quoted incorrectly by Homer.

Trivia 2:

In the year  2000, Uganda issued a 2000 shilling coin in the shape of a right-angled triangle. It had an image of Pythagoras and a statement of his theorem on one side.

Trivia 3:

In 1940 Elisha Scott Loomis published a book containing 256 different proofs of Pythagoras’ theorem. One of them was written by James Garfield, president of the United States, in 1881.

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