The truncated octahedron is a space-filling polyhedron. The truncated octahedron is an Archimedean polyhedron and you can think it as a octahedron from which you have removed six tetrahedra. Here we can see another point of view: A truncated octahedron is built by eight half cubes. This relation between the cube and the truncated octahedron can help us to understand the space-filling property of this beautiful polyhedron.

We have already calculated the volume of half a cube. Then, the volume of a truncated octahedron is eight times the volume of half a cube:

You can play with the transparency in this mathlet version

LINKS

	Hexagonal section of a cube We can cut in half a cube by a plane and get a section that is a regular hexagon. Using eight of this pieces we can made a truncated octahedron.
	The truncated octahedron is a space-filling polyhedron These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.
	The volume of a truncated octahedron The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces and 6 square faces. Its volume can be calculated knowing the volume of an octahedron.
	Volume of an octahedron The volume of an octahedron is four times the volume of a tetrahedron. It is easy to calculate and then we can get the volume of a tetrahedron.
	The volume of the tetrahedron The volume of a tetrahedron is one third of the prism that contains it.
	Regular dodecahedron One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.