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Pythagoras' Theorem

Years ago, a man named Pythagoras found an amazing fact about triangles:

 

If the triangle had a right angle (90°) ...

... and you made a square on each of the three sides, then ...

... the biggest square had the exact same area as the other two squares put together!

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.


So, the square of a (a²) plus the square of b (b²) is equal to the square of c (c²):

a2 + b2 = c2

Sure ... ?

Let's see if it really works using an example. A "3,4,5" triangle has a right angle in it, so the formula should work.

pythagoras theorem
Let's check if the areas are the same:
32 + 42 = 52

Calculating this becomes:
9 + 16 = 25


yes, it works !

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, then Pythagoras' Theorem allows us to find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:
abc triangle a2 + b2 = c2

Now you can use algebra to find any missing value, as in the following examples:

right angled triangle

a2 + b2 = c2

52 + 122 = c2

25 + 144 = 169

c2 = 169

c = √169

c = 13

right angled triangle

a2 + b2 = c2

92 + b2 = 152

81 + b2 = 225

Take 81 from both sides

b2 = 144

b = √144

b = 12


And You Can Prove It Yourself !

Get paper pen and scissors, then using the following animation as a guide:
  • Draw a right angled triangle on the paper, leaving plenty of space.
  • Draw a square along the hypotenuse (the longest side)
  • Draw the same sized square on the other side of the hypotenuse
  • Draw lines as shown on the animation, like this:
  • cut sqaure
  • Cut out the shapes
  • Arrange them so that you can prove that the big square has the same area as the two squares on the other sides

Another, Amazingly Simple, Proof

Here is one of the oldest proofs that the square on the long side has the same area as the other squares.

Watch the animation, and pay attention when the triangles start sliding around.

You may want to watch the animation a few times to understand what is happening.

The purple triangle is the important one.

before after

 

We also have a proof by adding up the areas.

history Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived !