A cuboctahedron is an Archimedean solid. It is generated by truncating the vertices of a cube or of an octahedron at 1/2 edge-length. The cube and the octahedron are dual polyhedra. There are 6 square faces on the cuboctahedron, one for each vertex of the octahedron. There are 8 equilateral triangular faces, one for each face of the octahedron.

We are going to calculate the volume of an octahedron of edge-length 1 starting from the volume of an octahedron.

If a cuboctahedron has edge-length 1, the octahedron that contains it is:

The volume of an octahedron of edge-length 2 is:

To calculate the volume of a cuboctahedron we have to subtract from the volume of the octahedron the volume of the 6 pyramids that we cut off.

Each pyramid is half octahedron. The volume of one of these pyramids is:

Now, we can calculate the volume of the cuboctahedron (what we subtract, 6 pyramids, may be reasssembled into three octahedron of edge-length 1)

Then the volume of a cuboctahedron of edge-length a is:

LINKS

	Leonardo da Vinci: Drawing of a cuboctahedron made to Luca Pacioli's De divina proportione. Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the cuboctahedron.
	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.
	Stellated cuboctahedron The compound polyhedron of a cube and an octahedron is an stellated cuboctahedron.It is the same to say that the cuboctahedron is the solid common to the cube and the octahedron in this polyhedron.
	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.
	Leonardo da Vinci: Drawing of a truncated octahedron made to Luca Pacioli's De divina proportione. Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the truncated octahedron.
	Leonardo da Vinci: Drawing of a dodecahedron made to Luca Pacioli's De divina proportione. Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the dodecahedron.
	Leonardo da Vinci:Drawing of an stellated octahedron (stella octangula) made to Luca Pacioli's De divina proportione. Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the stellated octahedron (stella octangula).