# Triakis tetrahedron

In geometry, a triakis tetrahedron (or kistetrahedron[1]) is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the truncated tetrahedron.

Triakis tetrahedron

TypeCatalan solid
Coxeter diagram
Conway notationkT
Face typeV3.6.6
isosceles triangle
Faces12
Edges18
Vertices8
Vertices by type4{3}+4{6}
Symmetry groupTd, A3, [3,3], (*332)
Rotation groupT, [3,3]+, (332)
Dihedral angle129°31′16″
arccos(−7/11)
Propertiesconvex, face-transitive

Truncated tetrahedron
(dual polyhedron)

Net

The triakis tetrahedron can be seen as a tetrahedron with a triangular pyramid added to each face; that is, it is the Kleetope of the tetrahedron. It is very similar to the net for the 5-cell, as the net for a tetrahedron is a triangle with other triangles added to each edge, the net for the 5-cell a tetrahedron with pyramids attached to each face. This interpretation is expressed in the name.

The length of the shorter edges is 3/5 that of the longer edges[2]. If the triakis tetrahedron has shorter edge length 1, it has area 5/311 and volume 25/362.

## Tetartoid symmetry

The triakis tetrahedron can be made as a degenerate limit of a tetartoid:

## Orthogonal projections

Orthogonal projection
Centered by Edge normal Face normal Face/vertex Edge
Triakis
tetrahedron
(Dual)
Truncated
tetrahedron
Projective
symmetry
[1] [1] [3] [4]

## Variations

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

If the triangles are right-angled isosceles, the faces will be coplanar and form a cubic volume. This can be seen by adding the 6 edges of tetrahedron inside of a cube.

## Stellations

This chiral figure is one of thirteen stellations allowed by Miller's rules.

The triakis tetrahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.