Truncated Tetrahedron

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The truncated tetrahedron is the Archimedean solid with faces . It is also the uniform polyhedron with Maeder index 2 (Maeder 1997), Wenninger index 6 (Wenninger 1989), Coxeter index 16 (Coxeter et al. 1954), and Har'El index 7 (Har'El 1993). It has Schläfli symbol t and Wythoff symbol . It is illustrated above together with a wireframe version and a net that can be used for its construction.

Some symmetric projections of the truncated tetrahedron are illustrated above.

It is implemented in the Wolfram Language as PolyhedronData["TruncatedTetrahedron"] or UniformPolyhedron["TruncatedTetrahedron"]. Precomputed properties are available as PolyhedronData["TruncatedTetrahedron"].

The skeleton of the truncated tetrahedron is the truncated tetrahedral graph, illustrated above in a number of embeddings.

The dual polyhedron of the truncated tetrahedron is the triakis tetrahedron, both of which are illustrated above together with their common midsphere. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are

			(1)
			(2)
			(3)

The distances from the center of the solid to the centroids of the triangular and hexagonal faces are given by

			(4)
			(5)

The surface area and volume are

			(6)
			(7)

The unit truncated tetrahedron has Dehn invariant

			(8)
			(9)
			(10)

(OEIS A377277), where the first expression uses the basis of Conway et al. (1999).