Pythagorean theorem

c is the longest side of the triangle a & b are the other two sides


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 tothe sum of the squares of the other two sides.

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Example: A "3, 4, 5" triangle has a right angle in it.


Let"s check if the areas are the same:

32 + 42 = 52

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 bởi vì I Use it?

Write it down as an equation:

a2 + b2 = c2

Example: Solve this triangle



Read Builder"s Mathematics to see practical uses for this.

Also read about Squares and Square Roots to lớn find out why √169 = 13

Example: Solve this triangle.


Example: What is the diagonal distance across a square of form size 1?


It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2, then the triangle is right angled.

Example: Does this triangle have a Right Angle?

Does a2 + b2 = c2 ?

a2 + b2 = 102 + 242 = 100 + 576 = 676
c2 = 262 = 676

They are equal, so ...

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Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?

Does 82 + 152 = 162 ?

82 + 152 = 64 + 225 = 289, but 162 = 256

So, NO, it does not have a Right Angle

Example: Does this triangle have a Right Angle?


Does a2 + b2 = c2 ?

And You Can Prove The Theorem 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 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 lớn watch the animation a few times to understand what is happening.

The purple triangle is the important one.

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We also have a proof by adding up the areas.

Historical Note: while we điện thoại tư vấn it Pythagoras" Theorem, it was also known by Indian, Greek, Chinese & Babylonian mathematicians well before he lived.
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