Calculations at a triakis tetrahedron, the dual body of the truncated tetrahedron. A triakis tetrahedron is a regular tetrahedron with matching regular triangular pyramids attached to its faces. It has four vertices with three edges and four vertices with six edges. Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:
b = 3/5 * a
h = 3/5 * a * √6
A = 3/5 * a² * √11
V = 3/20 * a³ * √2
r_{m} = a/4 * √2
r_{i} = 3/4 * a * √ 2/11
A/V = 4 * √11 / ( a * √2)

The triakis tetrahedron is a Catalan solid. Lengths, height and radiuses have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^{-1}.

The Catalan solids were discovered in the nineteenth century by Eugène Charles Catalan. They are also called dual-Archimedean solids because they are dual to the Archimedean solids. A dual solid is created when the midpoints of the side faces of a polyhedron are connected to each other when these side faces are connected. Then the previous side faces are removed. In simple terms, the faces are swapped with the vertices and vice versa. The dual solid of the dual solid is again the original polyhedron. The triakis tetrahedron is the first Catalan solid because it has the fewest number of vertices. Since there are thirteen Archimedean solids, there are also this number of Catalan solids. The side faces of the Catalan solids are each identical polyhedra, but they are not perfectly regular. This is in contrast to the Archimedean solids, where the side faces are different but perfectly regular. Catalan solids all have an insphere and a midsphere. The insphere touches all faces, the midsphere touches all edges. They do not have a circumsphere that would touch all vertices.