Cube Face Diagonal

Every cube carries two different diagonals, and mixing them up is the single most common error in solid geometry. The face diagonal runs corner to corner across one square face and stays on the surface, giving $d = s\sqrt{2} \approx 1.414s$. The space diagonal cuts through the interior from one corner to the far opposite corner and is longer, $s\sqrt{3} \approx 1.732s$. For a 1 m cube those are 1.41 m and 1.73 m, a 22 percent gap that will ruin a cut list.

The face diagonal is just the square's diagonal, since a cube face is a square, so it inherits $\sqrt{2}$ and everything that comes with it — including the fact that this was the first number ever proved irrational, some time in the 5th century BC. Practically, it sets the length of a diagonal brace across one side of a cubic frame, and the largest square opening a cube-shaped object can be turned through.