Space diagonal

In geometry a space diagonal (also interior diagonal, body diagonal or triagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with face diagonals, which connect vertices on the same face (but not on the same edge) as each other. The word "triagonal" is derived from the fact that as a variable point travels down the line, three coordinates change. The equivalent concept in a square or other polygon is a diagonal, because two coordinates change. In a tesseract the equivalent is a quadragonal because four coordinates change, etc.

For example, a tetrahedron has no space diagonals, while a cube (shown at right) or more generally a parallelepiped has four space diagonals.

An axial diagonal is a space diagonal that passes through the center of a polyhedron.

For example, in a cube with edge length a, all four space diagonals are axial diagonals, of common length ${\displaystyle a{\sqrt {3}}.}$ More generally, a cuboid with edge lengths a, b, and c has all four space diagonals axial, with common length ${\displaystyle {\sqrt {a^{2}+b^{2}+c^{2}}}.}$

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