Pythagoras' Theorem
Over 2000 years ago there was an amazing discovery about triangles: When the triangle has a right angle (90°) ... ... and squares are made on each
of the 

... the biggest square has the exact same area as the other two squares put together! 
It is called "Pythagoras' Theorem" and can be written in one short equation:
a^{2} + b^{2} = c^{2}
Note:
 c is the longest side of the triangle
 a and b are the other two sides
Definition
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.
Sure ... ?
Let's see if it really works using an example.
Example: A "3,4,5" triangle has a right angle in it.
Let's check if the areas are the same: 3^{2} + 4^{2} = 5^{2} Calculating this becomes: 9 + 16 = 25 It works ... like Magic! 
Why Is This Useful?
If we know the lengths of two sides of a right angled triangle, we can 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:
a^{2} + b^{2} = c^{2} 
Now you can use algebra to find any missing value, as in the following examples:
Example: Solve this triangle.
a^{2} + b^{2} = c^{2} 5^{2} + 12^{2} = c^{2} 25 + 144 = c^{2} 169 = c^{2} c^{2} = 169 c = √169 c = 13 
You can also read about Squares and Square Roots to find out why √169 = 13
Example: Solve this triangle.
a^{2} + b^{2} = c^{2} 9^{2} + b^{2} = 15^{2} 81 + b^{2} = 225 Take 81 from both sides: b^{2} = 144 b = √144 b = 12 
Example: What is the diagonal distance across a square of size 1?
a^{2} + b^{2} = c^{2} 1^{2} + 1^{2} = c^{2} 1 + 1 = c^{2} 2 = c^{2} c^{2} = 2 c = √2 = 1.4142... 
It works the other way around, too: when the three sides of a triangle make a^{2} + b^{2} = c^{2}, then the triangle is right angled.
Example: Does this triangle have a Right Angle?
Does a^{2} + b^{2} = c^{2} ?
They are equal, so ... Yes^{}, it does have a Right Angle! 
Example: Does an 8, 15, 16 triangle have a Right Angle?
Does 8^{2} + 15^{2} = 16^{2 }?
 8^{2} + 15^{2} = 64 + 225 = 289,
 but 16^{2 }= 256
So, NO, it does not have a Right Angle
Example: Does this triangle have a Right Angle?
Does a^{2} + b^{2} = c^{2} ? Does (√3)^{2} + (√5)^{2} = (√8)^{2} ?
Does 3 + 5 = 8 ?
Yes^{}, it does! So this is a rightangled triangle 
And You Can Prove The Theorem Yourself !
Get paper pen and scissors, then using the following animation as a guide:

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.
becomes 
We also have a proof by adding up the areas.
Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived ! 