The experience of building an octahedron with paper or using 12 plastic tubes forming 4 squares in three orthogonal planes, or just piling 6 oranges is not replaced with a two dimensional representation.

An interactive application can remind us how can we take a look at a octahedron.

An octahedron is composed by two pyramids of square base. We can see the height of these two pyramides as the diagonal of a square.

The diagonal of a square of edge length 1 is:

Therefore, the volume of an octahedron of edge length 1 is:

And the volume of an octahedron of edge length a is:

Using that we can calculate the volume of a tetrahedron. We can consider a tetrahedron of edge length 2:

We can write a relation:

A tetrahedron of edge length 2 is made of one octahedron and four tetrahedra of edge length 1:

Then, the volume of an octahedron is four times the volume of a tetrahedron and we can recalculate the volume of a tetrahedron.

LINKS

	The volume of the tetrahedron The volume of a tetrahedron is one third of the prism that contains it.
	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.
	The volume of an stellated octahedron (stella octangula) The stellated octahedron was drawn by Leonardo for Luca Pacioli's book 'De Divina Proportione'. A hundred years later, Kepler named it stella octangula.
	The volume of a cuboctahedron A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of a cube.
	The volume of a cuboctahedron (II) A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an 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 icosahedron and its volume The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
	Regular dodecahedron One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.