What is the square of right angle triangle?

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!

It is called "Pythagoras' 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.

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 Do I Use it?

Write it down as an equation:

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

Start with:a2 + b2 = c2

Put in what we know:52 + 122 = c2

Calculate squares:25 + 144 = c2

25+144=169:169 = c2

Swap sides:c2 = 169

Square root of both sides:c = √169

Calculate:c = 13

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

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

Start with:a2 + b2 = c2

Put in what we know:92 + b2 = 152

Calculate squares:81 + b2 = 225

Take 81 from both sides: 81 − 81 + b2 = 225 − 81

Calculate: b2 = 144

Square root of both sides:b = √144

Calculate:b = 12

Start with:a2 + b2 = c2

Put in what we know:12 + 12 = c2

Calculate squares:1 + 1 = c2

1+1=2: 2 = c2

Swap sides: c2 = 2

Square root of both sides:c = √2

Which is about:c = 1.4142...

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

Does a2 + b2 = c2 ?

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

They are equal, so ...

Yes, it does 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

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

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 Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived.

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Activity: Pythagoras' Theorem
Activity: A Walk in the Desert

A right-angled triangle (also called a right triangle) is a triangle with a right angle (90°) in it.

The little square in the corner tells us it is a right angled triangle (I also put 90°, but you don't need to!)

The right angled triangle is one of the most useful shapes in all of mathematics!
(It is used in the Pythagoras Theorem and Sine, Cosine and Tangent for example).

Try it yourself (drag the points):

geometry/images/triangle.js?mode=right

There are two types of right angled triangle:

Isosceles right-angled triangle

One right angle
Two other equal angles always of 45°
Two equal sides

Scalene right-angled triangle

One right angle
Two other unequal angles
No equal sides

The "3,4,5 Triangle" has a right angle in it.

(Draw one if you ever need a right angle!)

It has no equal sides so it is a scalene right-angled triangle

And, like all triangles, the three angles always add up to 180°.

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