Augmented truncated tetrahedron
Type	Johnson J64 – J65 – J66
Faces	8 triangles 3 squares 3 hexagons
Edges	27
Vertices	15
Vertex configuration	2x3(3.62) 3(3.4.3.4) 6(3.4.3.6)
Symmetry group	C3v
Properties	convex
Net

In geometry, the augmented truncated tetrahedron is a polyhedron constructed by attaching a triangular cupola onto an truncated tetrahedron. It is an example of a Johnson solid.

The augmented truncated tetrahedron is constructed from a truncated tetrahedron by attaching a triangular cupola.^[1] This cupola covers one of the truncated tetrahedron's four hexagonal faces, so that the resulting polyhedron's faces are eight equilateral triangles, three squares, and three regular hexagons.^[2] Since it has the property of convexity and has regular polygonal faces, the augmented truncated tetrahedron is a Johnson solid, denoted as the sixty-fifth Johnson solid ${\displaystyle J_{65}}$.^[3]

The surface area of an augmented truncated tetrahedron is:^[2] $${\displaystyle {\frac {6+13{\sqrt {3}}}{2}}a^{2}\approx 14.258a^{2},}$$ the sum of the areas of its faces. Its volume can be calculated by slicing it off into both truncated tetrahedron and triangular cupola, and adding their volume:^[2] $${\displaystyle {\frac {11{\sqrt {2}}}{4}}a^{3}\approx 3.889a^{3}.}$$

It has the same three-dimensional symmetry group as the triangular cupola, the pyramidal symmetry ${\displaystyle C_{3\mathrm {v} }}$. Its dihedral angles can be obtained by adding the angle of a triangular cupola and an augmented truncated tetrahedron in the following:^[4]

• its dihedral angle between triangle and hexagon is as in the truncated tetrahedron: 109.47°;
• its dihedral angle between adjacent hexagons is as in the truncated tetrahedron: 70.53°;
• its dihedral angle between triangle and square is as in the triangular cupola's angle: 125.3°
• its dihedral angle between triangle and square, on the edge where the triangular cupola and truncated tetrahedron are attached, is the sum of both triangular cupola's square-hexagon angle and the truncated tetrahedron's triangle-hexagon angle: approximately 164.17°; and
• its dihedral angle between triangle and hexagon, on the edge where triangular cupola and truncated tetrahedron are attached, is the sum of the dihedral angle of a triangular cupola and truncated tetrahedron between that: approximately 141.3°;

• Weisstein, Eric W., "Augmented truncated tetrahedron" ("Johnson solid") at MathWorld.

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