In geometry, the truncated tetrahedron is an Archimedean solid.

It can be constructed by truncating all 4 vertices of a regular tetrahedron.

[1] The resulting polyhedron has 4 equilateral triangles and 4 regular hexagons, 18 edges, and 12 vertices.

[2] With edge length 1, the Cartesian coordinates of the 12 vertices are points

is the sum of 4 regular hexagons and 4 equilateral triangles' area, and its volume

[3] The densest packing of the truncated tetrahedron is believed to be

, as reported by two independent groups using Monte Carlo methods by Damasceno, Engel & Glotzer (2012) and Jiao & Torquato (2013) harvtxt error: no target: CITEREFJiaoTorquato2013 (help).

[4][5] Although no mathematical proof exists that this is the best possible packing for the truncated tetrahedron, the high proximity to the unity and independence of the findings make it unlikely that an even denser packing is to be found.

[4] The truncated tetrahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.

[8] The truncated tetrahedron can be found in the construction of polyhedrons.

For example, the augmented truncated tetrahedron is a Johnson solid constructed from a truncated tetrahedron by attaching triangular cupola onto its hexagonal face.

It is classified as plesiohedron, meaning it can tessellate in three-dimensional space known as honeycomb; an example is triakis truncated tetrahedral honeycomb.

B. Friauf in which he described it as a intermetallic structure formed by a compound of metallic elements.

[11] It can be found in crystals such as complex metallic alloys, an example is dizinc magnesium MgZn2.

[12] It is a lower symmetry version of the truncated tetrahedron, interpreted as a truncated tetragonal disphenoid with its three-dimensional symmetry group as the dihedral group

[citation needed] Truncating a truncated tetrahedron gives the resulting polyhedron 54 edges, 32 vertices, and 20 faces—4 hexagons, 4 nonagons, and 12 trapeziums.

This polyhedron was used by Adidas as the underlying geometry of the Jabulani ball designed for the 2010 World Cup.