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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 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 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.

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.

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

Volume of a cuboctahedron: Cuboctahedron inside an octahedron | matematicasvisuales

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:

Volume of a cuboctahedron: the volume of a pyramid that we cut off from the octahedron  | matematicasvisuales

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

Volume of a cuboctahedron The volume of a cuboctahedron is the volume of an octahedron of edge 2 minus three octahedron of edge 1| matematicasvisuales
Volume of a cuboctahedron: Cuboctahedron | matematicasvisuales

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

Volume of a cuboctahedron Cuboctahedron made with six business cards (Origami Resource Center)| matematicasvisuales
Origami cuboctahedron made with six business cards (Instructions in Origami Resource Center, click over the image to get the instructions)

Cuboctahedron and octahedron made with rubber bands and paper: (Go to Resources: Polyhedra made with rubber bands and paper if you want to build polyhedra using this technique.)

Volume of a cuboctahedron: a cuboctahedron made with rubber bands and paper | matematicasvisuales
Volume of a cuboctahedron: a cuboctahedron and a octahedron made with rubber bands and paper | matematicasvisuales
Volume of a cuboctahedron: plane net of a cuboctahedron made with rubber bands and paper | matematicasvisuales

It was Durer the first to publish plane nets of polyhedra. In his book 'Underweysung der Messung' ('Four Books of Measurement', published in 1525) the author draw plane developments of several Platonic and Archimedean solids, for example, this cuboctahedron:

Volume of a cuboctahedron: plane net of a cuboctahedron drawn by Durer | matematicasvisuales

REFERENCES

Hugo Steinhaus - Mathematical Snapshots - Oxford University Press - Third Edition (p. 197)
Magnus Wenninger - 'Polyhedron Models', Cambridge University Press.
Peter R. Cromwell - 'Polyhedra', Cambridge University Press, 1999.
H.Martin Cundy and A.P. Rollet, 'Mathematical Models', Oxford University Press, Second Edition, 1961 (p. 87).
W.W. Rouse Ball and H.S.M. Coxeter - 'Matematical Recreations & Essays', The MacMillan Company, 1947.
Luca Pacioli - De divina proportione - (La divina proporción) Ediciones Akal, 4ª edición, 2004. Spanish translation by Juan Calatrava.

MORE LINKS

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 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 the tetrahedron
The volume of a tetrahedron is one third of the prism that contains it.
Plane developments of geometric bodies: Tetrahedron
The first drawing of a plane net of a regular tetrahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Plane developments of geometric bodies (1): Nets of prisms
We study different prisms and we can see how they develop into a plane net. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (2): Prisms cut by an oblique plane
Plane nets of prisms with a regular base with different side number cut by an oblique plane.
Plane developments of geometric bodies (3): Cylinders
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (4): Cylinders cut by an oblique plane
We study different cylinders cut by an oblique plane. The section that we get is an ellipse.
Plane developments of geometric bodies (5): Pyramid and pyramidal frustrum
Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.
Plane developments of geometric bodies (7): Cone and conical frustrum
Plane developments of cones and conical frustum. How to calculate the lateral surface area.
Plane developments of geometric bodies (8): Cones cut by an oblique plane
Plane developments of cones cut by an oblique plane. The section is an ellipse.
Plane developments of geometric bodies: Dodecahedron
The first drawing of a plane net of a regular dodecahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Resources: How to build polyhedra using paper and rubber bands
A very simple technique to build complex and colorful polyhedra.
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
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
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.
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.
A truncated octahedron made by eight half cubes
Using eight half cubes we can make a truncated octahedron. The cube tesselate the space an so do the truncated octahedron. We can calculate the volume of 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.
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 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.
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).
Truncated tetrahedron
The truncated tetrahedron is an Archimedean solid made by 4 triangles and 4 hexagons.
Truncations of the cube and octahedron
When you truncate a cube you get a truncated cube and a cuboctahedron. If you truncate an octahedron you get a truncated octahedron and a cuboctahedron.
Chamfered Cube
You can chamfer a cube and then you get a polyhedron similar (but not equal) to a truncated octahedron. You can get also a rhombic dodecahedron.