Editing Augmented truncated tetrahedron (section)

== Properties ==
The surface area of an augmented truncated tetrahedron is:{{r|berman}}
<math display="block"> \frac{6 + 13 \sqrt{3}}{2}a^2 \approx 14.258a^2, </math>
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:{{r|berman}}
<math display="block"> \frac{11 \sqrt{2}}{4}a^3 \approx 3.889a^3. </math>

It has the same [[Point groups in three dimensions|three-dimensional symmetry group]] as the triangular cupola, the [[pyramidal symmetry]] <math> C_{3 \mathrm{v}} </math>. Its [[dihedral angle]]s can be obtained by adding the angle of a triangular cupola and an augmented truncated tetrahedron in the following:{{r|johnson}}
* 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°;