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:

a2 + b2 = c2 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:

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

Pythagoras tells us that c = √(x2 + y2) Now we make another triangle with its base along the "√(x2 + y2)" side of the previous triangle, and going up to the far corner: We can use Pythagoras again, but this time the two sides are √(x2 + y2) and z, and we get this formula: And the final result is: So it is all part of a pattern that extends onwards:

DimensionsPythagorasDistance "c"
1c2 = x2√(x2) = x
2c2 = x2 + y2√(x2 + y2)
3c2 = x2 + y2 + z2√(x2 + y2 + z2)
.........
nc2 = a12 + a22 + ... + an2√(a12 + a22 + ... + an2)

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