Pythagorean Theorem in 3D

Portrait of the mathematician Pythagoras
Pythagoras

In 2D

First, let's 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!


Right triangle with sides labeled a, b, and hypotenuse c

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

a2 + b2 = c2

Visual equation showing squares on sides a and b equal to the square on side c

Note:

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 Pythagorean 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:

Cuboid with a diagonal line connecting opposite corners through the center

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

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

Cuboid highlighting the diagonal of the base rectangle

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

Cuboid showing an internal vertical triangle formed by the base diagonal and the height

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

Formula: distance equals the square root of x squared plus y squared plus z squared

And the final result is:

pythagoras 3d

Cuboid with a diagonal line connecting opposite corners through the center

Example: What's the distance from one corner of a room to the opposite corner if the room is 3 meters wide, 4 meters deep, and 2.4 meters high?

  1. First, find the diagonal along the floor:
    √(32 + 42) = √(9 + 16) = √25 = 5 m
  2. Now use that 5m length and the 2.4m height to find the main diagonal:
    √(52 + 2.42) = √(25 + 5.76) = √30.76 ≈ 5.546 m

The distance is about 5.546 m.

Or in one go:

√(32 + 42 + 2.42) = √(9 + 16 + 5.76) = √30.76 ≈ 5.546 m

Pattern

It is all part of a pattern that extends onwards:

Dimensions Pythagoras Distance "c"
1 (a line) 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 we have a formula for calculating distance in any number of dimensions!

Pythagorean Theorem Right Angled Triangles The Fishing Rod Pythagorean Theorem Algebra Proof