Pythagorean+Theorum

=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
 * [[image:http://www.mathsisfun.com/geometry/images/triangle-abc.gif width="180" height="100"]] ||

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. 32 + 42 = 52 Calculating this becomes: 9 + 16 = 25
 * [[image:http://www.mathsisfun.com/images/pythagoras.gif width="154" height="170" caption="pythagoras theorem"]] || Let's check if the areas **are** the same:

//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:
 * [[image:http://www.mathsisfun.com/geometry/images/triangle-abc.gif width="180" height="100" caption="abc triangle"]] || a2 + b2 = c2 ||

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

a2 + b2 = c2 52 + 122 = c2 25 + 144 = 169 c2 = 169 c = √169 c = 13 a2 + b2 = c2 92 + b2 = 152 81 + b2 = 225 Take 81 from both sides b2 = 144 b = √144 b = 12
 * [[image:http://www.mathsisfun.com/images/triangle4.gif width="186" height="107" caption="right angled triangle"]]
 * [[image:http://www.mathsisfun.com/images/triangle3.gif width="186" height="107" caption="right angled triangle"]]

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:
 * [[image:http://www.mathsisfun.com/images/pythagoras-cutout.png width="102" height="101" align="top" caption="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
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.
 * Here is one of the oldest proofs that the square on the long side has the same area as the other squares. ||
 * [[image:http://www.mathsisfun.com/geometry/images/pythagoras-proof-2a.gif width="140" height="140" align="top" caption="before"]] || [[image:http://www.mathsisfun.com/geometry/images/pythagoras-proof-2b.gif width="140" height="140" align="top" caption="after"]] || ||
 * [[image:http://www.mathsisfun.com/geometry/images/pythagoras-proof-2a.gif width="140" height="140" align="top" caption="before"]] || [[image:http://www.mathsisfun.com/geometry/images/pythagoras-proof-2b.gif width="140" height="140" align="top" caption="after"]] || ||

We also have a [|proof by adding up the areas].
 * [[image:http://www.mathsisfun.com/images/style/scroll.jpg width="48" height="48" caption="history"]] || //Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived !// ||