The cell-first perspective projection of the 16-cell into 3-space has a triakis tetrahedron envelope.

It is constructed from taking a triakis tetrahedron by truncating the order-6 vertices.

If the triakis tetrahedron has shorter edge lengths 1, it has area and volume .

A triakis tetrahedron with equilateral triangle faces represents a net of the four-dimensional regular polytope known as the 5-cell.

In geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid.

The triakis tetrahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane.

If all of a triakis tetrahedron's vertices, of both kinds, are truncated, the resulting solid is an irregular icosahedron, whose dual is a trihexakis truncated tetrahedron.

The truncated triakis tetrahedron is a convex polyhedron with 16 faces: 4 sets of 3 pentagons arranged in a tetrahedral arrangement, with 4 hexagons in the gaps.

The triakis tetrahedron is the Kleetope of a tetrahedron, the triakis octahedron is the Kleetope of an octahedron, and the triakis icosahedron is the Kleetope of an icosahedron.