Augmented truncated tetrahedron

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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.Template:R 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.Template:R 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 $J_{65}$.Template:R

Properties

The surface area of an augmented truncated tetrahedron is:Template:R $$\frac{6+13\sqrt{3}}{2}{a}^2\approx 14.258a^2,$$ 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:Template:R $$\frac{11\sqrt{2}}{4}{a}^3\approx 3.889a^3.$$

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

• 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°;

External links

• Template:Mathworld2

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