Augmented truncated tetrahedron

Augmented truncated tetrahedron
Type	Johnson J64 – J65 – J66
Faces	12 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

3D model of an augmented truncated tetrahedron

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.

Construction

The augmented truncated tetrahedron is constructed from a truncated tetrahedron by attaching a triangular cupola. This cupola covers the hexagonal face so that the resulting polyhedron has the face of twelve equilateral triangles, three squares, and 3 regular hexagons. 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 $J_{65}$.

Properties

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

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

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

External links

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