Augmented truncated tetrahedron

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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