Pythagoras Theorem - Math Is Fun

Pythagorean Theorem

Pythagoras

Over 2000 years ago there was an amazing discovery about triangles:

When a triangle has a right angle (90°) ...

... and squares are made on each of the three sides, ...

geometry/images/pyth1.js

... then the biggest square has the exact same area as the other two squares put together! (press Go).

It is the "Pythagorean Theorem" and can be written in one short equation:

a2 + b2 = c2

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: 32 + 42 = 52 Calculating this becomes: 9 + 16 = 25 It works ... like Magic!

Why Is This Useful?

When we know two side lengths of a right triangle we can find the third side length.

How Do I Use it?

Write it down as an equation:

a2 + b2 = c2

Then we use algebra to find any missing value, as in these examples:

Example: Solve this triangle

Start with:a2 + b2 = c2Put in what we know:52 + 122 = c2Calculate squares:25 + 144 = c225+144=169:169 = c2Swap sides:c2 = 169Square root of both sides:c = √169 Calculate:c = 13

Read about Squares and Square Roots to discover why √169 = 13

Example: Solve this triangle.

Start with:a2 + b2 = c2Put in what we know:92 + b2 = 152Calculate squares:81 + b2 = 225 Take 81 from both sides: 81 − 81 + b2 = 225 − 81Calculate: b2 = 144Square root of both sides:b = √144 Calculate:b = 12

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

Start with:a2 + b2 = c2Put in what we know:12 + 12 = c2Calculate squares:1 + 1 = c21+1=2: 2 = c2Swap sides: c2 = 2Square root of both sides:c = √2Which is about:c = 1.4142... There are practical uses! Read Builder's Mathematics.

Is it a Right Angle?

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

Which shows us that a2 + b2 is equal to c2, so ...

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 = 289 162 = 256

they are not equal

So, NO, it does not have a Right Angle

Example: Does this triangle have a Right Angle?

Does a2 + b2 = c2 ? Does (√3)2 + (√5)2 = (√8)2 ? Does 3 + 5 = 8 ?

Yes, it does!

So this is a right-angled triangle

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 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 Pythagorean Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived. 511,512,617,618, 1145, 1146, 1147, 2359, 2360, 2361 Activity: Pythagorean Theorem Activity: A Walk in the Desert Right Angled Triangles The Fishing Rod Pythagorean Theorem in 3D Pythagorean Generalizations Pythagorean Areas Triangles Pythagorean Triples Pythagorean Theorem Algebra Proof