In
geometry, a face diagonal of a
polyhedron is a
diagonal on one of the
faces, in contrast to a ''
space diagonal'' passing through the interior of the polyhedron.
A
cuboid
In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...
has twelve face diagonals (two on each of the six faces), and it has four space diagonals. The cuboid's face diagonals can have up to three different lengths, since the faces come in congruent pairs and the two diagonals on any face are equal. The cuboid's space diagonals all have the same length. If the edge lengths of a cuboid are ''a'', ''b'', and ''c'', then the distinct rectangular faces have edges (''a'', ''b''), (''a'', ''c''), and (''b'', ''c''); so the respective face diagonals have lengths
and
Thus each face diagonal of a
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
with side length ''a'' is
.
A regular
dodecahedron has 60 face diagonals (and 100 space diagonals).
[.]
References
Elementary geometry
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