# Pythagoras' Theorem in 3D

## In 2D

First, let us have a quick refresher in two dimensions:

*Pythagoras*

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

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

*... 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:

c^{2} = a^{2} + b^{2}

Note:

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

And when we want to know the distance "c" we take the square root:

c = √(a^{2} + b^{2})

You can read more about it at Pythagoras' Theorem, but here we see how it can be extended into **3 Dimensions**.

## In 3D

Let's say we want the distance from the bottom-most left front corner to the top-most right back corner of this cuboid:

First let's just do the triangle on the bottom.

Pythagoras tells us that **c = √(x ^{2} + y^{2})**

Now we make another triangle with its base along the "**√(x ^{2} + y^{2})**" side of the previous triangle, and going up to the far corner:

We can use Pythagoras again, but this time the two sides are ** √(x ^{2} + y^{2})** and

**z**, and we get this formula:

And the final result is:

So it is all part of a pattern that extends onwards:

Dimensions | Pythagoras | Distance "c" |
---|---|---|

1 |
c^{2} = x^{2} |
√(x^{2}) = x |

2 |
c^{2} = x^{2} + y^{2} |
√(x^{2} + y^{2}) |

3 |
c^{2} = x^{2} + y^{2} + z^{2} |
√(x^{2} + y^{2} + z^{2}) |

... | ... | ... |

n |
c^{2} = a_{1}^{2} + a_{2}^{2} + ... + a_{n}^{2} |
√(a_{1}^{2} + a_{2}^{2} + ... + a_{n}^{2}) |

So next time you need an n-dimensional distance you will know how to calculate it!