An augmented tetrahedron is an augmented polyhedron obtained by augmenting the four faces of a tetrahedron with four tetrahedra, leading to a concave polyhedron with 8 vertices, 18 edges, and 12 faces. It is therefore a (nonregular, concave) dodecahedron.
When the base tetrahedron is a regular tetrahedron, the resulting equilateral polyhedron is a concave deltahedron, illustrated above and implemented in the Wolfram Language as PolyhedronData["EquilateralAugmentedTetrahedron"]. For a base tetrahedron with edge lengths , the resulting polyhedron has surface area and volume given by
(1)
 
(2)

The convex hull of this polyhedron is the triakis tetrahedron.