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What is the relation between the length of the space diagonal \(a\) and the lengths of the edges \(x\), \(y\) and \(z\) in a three-dimensional box?

Figure 1: Space diagonal \(a\) in a box with sides \(x\), \(y\) and \(z\).

If we draw the face diagonal \(b\) for the rectangle defined by the edges \(x\) and \(y\), we can see that we have a right triangle (blue below) with the sides \(x\), \(y\) and the hypotenuse \(b\).

Figure 2: Space diagonal \(a\) and face diagonal \(b\) in a box, where \(b^2 = x^2 + y^2\).

Now we can see that the space diagonal \(a\) is the hypotenuse of a right triangle (red below), with the sides \(b\) and \(z\).

Figure 3: Space diagonal \(a\) and face diagonal \(b\) in a box, where \(a^2 = b^2 + z^2\).

Thanks to Pythagoras, we have these relations between the variables

`b^2 = x^2 + y^2`
`a^2 = b^2 + z^2 = x^2 + y^2 + z^2`
`a = \sqrt{x^2 + y^2 + z^2}`

So the length of the space diagonal in a box is expressed by a simple equation between the length of the three edges.

Diagonal in a cube

In the special case of a cube all edges are equal (\(x\) = \(y\) = \(z\))

`a^2 = x^2 + x^2 + x^2 = 3x^2`
`a = \sqrt{3x^2} = x\sqrt{3}`

Note: If you want to bring your 165 cm skis when traveling, but the maximum package dimension for a side is 100 cm, you can solve your problem by putting the skis in a 100x100x100 cm box.

Diagonal in a square

We leave the three-dimensional geometry and examine the two-dimension instead. For a square with edges of length \(x\), the face diagonal \(b\) is

`b^2 = x^2 + x^2 = 2x^2`
`b = \sqrt{2x^2} = x\sqrt{2}`

Diagonal in a line

Now to the one-dimension. For a line of length \(x\), the "diagonal" \(c\) can be expressed as

`c^2 = x^2 = 1x^2`
`c = \sqrt{1x^2} = x\sqrt{1}`

Note: The length of the "diagonal" of a line is of course equal to the length of the line itself, but we want to follow the pattern from the previous equations.

Diagonal in an n-dimensional hypercube

It seems to be so nice, that for the \(n\)-dimensional hypercube with edges of length \(x\), the diagonal \(d\) is calculated by

`d = x\sqrt{n}`
First published by Anders Gustafson 2018-03-03