Cube Space Diagonal

The space diagonal runs through a cube's interior, corner to opposite corner, and it takes two applications of Pythagoras to reach. First across a face: that diagonal is $\sqrt{a^2 + a^2} = a\sqrt{2}$. Then the space diagonal is the hypotenuse of a right triangle whose legs are that face diagonal and the remaining edge, so $d = \sqrt{2a^2 + a^2} = a\sqrt{3} \approx 1.732a$. The relation is linear, so inverting it is a plain division with no branch to worry about.

The practical form of the question is whether a long thing will fit in a box. A 1 m cubic crate accepts a rod of up to 1.73 m; a 2.4 m cube-ish site container takes 4.16 m. The general version for a box that is not a cube is $d = \sqrt{l^2 + w^2 + h^2}$, the three-dimensional distance formula, which is worth knowing because most real boxes are not cubes — a van with a 3.2 m by 1.7 m by 1.4 m load space swallows a 3.9 m length on the diagonal that will not go in flat.

The pattern keeps going. In $n$ dimensions the diagonal of a unit hypercube is $\sqrt{n}$, which grows without limit while every edge stays exactly 1. In a thousand dimensions the corners are more than thirty units from each other across a box whose sides are all 1 — a genuinely strange fact, and one of the reasons high-dimensional data behaves so badly against intuition trained in three.

Two errors. The first is mixing up the two diagonals: the face diagonal is $a\sqrt{2} \approx 1.414a$, the space diagonal is $a\sqrt{3} \approx 1.732a$, and they are easy to confuse because both are "the diagonal" in ordinary speech. A quick check is that the space diagonal must be the longer of the two — it is the longest straight line the box contains. The second is a warning that this number is a ceiling and not a promise. Fitting the rod requires getting it in, which means passing it through an opening, and the limiting dimension there is usually the diagonal of the door or lid rather than the diagonal of the interior. A rod of exactly $a\sqrt{3}$ also has to arrive at precisely the right angle with zero clearance and zero thickness, none of which real objects have.