Pythagoras' Theorem in 3D

In 2D

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


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:

a2 + b2 = c2

pythagoras squares a^2 + b^2 = c^2


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

c2 = a2 + b2

c = √(a2 + b2)

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:

pythagoras 3d

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

Pythagoras tells us that c = √(x2 + y2)

pythagoras 3d

Now we make another triangle with its base along the "√(x2 + y2)" side of the previous triangle, and going up to the far corner:

pythagoras 3d

We can use Pythagoras again, but this time the two sides are √(x2 + y2) and z, and we get this formula:

pythagoras 3d

And the final result is:

pythagoras 3d


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

Dimensions Pythagoras Distance "c"
1 c2 = x2 √(x2) = x
2 c2 = x2 + y2 √(x2 + y2)
3 c2 = x2 + y2 + z2 √(x2 + y2 + z2)
... ... ...
n c2 = a12 + a22 + ... + an2 √(a12 + a22 + ... + an2)


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