Augmented triangular prism

Augmented triangular prism
Type	Johnson J48 – J49 – J50
Faces	6 triangles 2 squares
Edges	13
Vertices	7
Vertex configuration	$\begin{array}{cc}& 2\times (3\times 4^2)+\\ & 1\times (3^4)+\\ & 4\times (3^3\times 4)\end{array}$
Symmetry group	$C_{2\mathrm{v}}$
Dihedral angle (degrees)	triangle-triangle: 109.5°, 169.4° triangle-square: 90°, 114.7° square-square: 60°
Properties	convex, composite
Net

File:J49 augmented triangular prism.stl In geometry, the augmented triangular prism is a polyhedron constructed by attaching an equilateral square pyramid onto the square face of a triangular prism. As a result, it is an example of Johnson solid. It can be visualized as the chemical compound, known as capped trigonal prismatic molecular geometry.

Construction

The augmented triangular prism is composite: it can be constructed from a triangular prism by attaching an equilateral square pyramid to one of its square faces, a process known as augmentation. This square pyramid covers the square face of the prism, so the resulting polyhedron has six equilateral triangles and two squares as its faces. A convex polyhedron wherein all faces are polygonal regular is Johnson solid. The augmented triangular prism is among them, enumerated as the forty-ninth Johnson solid $J_{49}$.

Properties

An augmented triangular prism has a surface area $A$ by adding the area of six equilateral triangles and two squares. Its volume $V$ can be obtained by slicing it into a regular triangular prism and an equilateral square pyramid, and adding their volume subsequently. With edge length $a$, the formulations are: $$A=\frac{4+3\sqrt{3}}{2}{a}^2\approx 4.598a^2,V=\frac{2\sqrt{2}+3\sqrt{3}}{12}{a}^3\approx 0.669a^3.$$

The augmented triangular prism has three-dimensional symmetry group of the cyclic group $C_{2\mathrm{v}}$ of order four: its appearance is symmetrical by rotating a half around an axis of symmetry that passes through the pyramid's apex and base as well as a triangular prism's edge that is not attached by a pyramid, and mirroring two vertical mirror planes passing through any midpoint of attachment edges.

The dihedral angle (i.e., the angle between two polygonal faces) of an augmented triangular prism can be calculated by adding the angle of an equilateral square pyramid and a regular triangular prism in the following:

• The dihedral angle of an augmented triangular prism between two adjacent triangles is that of an equilateral square pyramid between two adjacent triangular faces, $\arccos (-1/3)\approx 109.5^{\circ }$
• The dihedral angle of an augmented triangular prism between two adjacent squares is that of a triangular prism between two lateral faces, the interior angle of a triangular prism $\pi /3=60^{\circ }$.
• The dihedral angle of an augmented triangular prism between the square and the triangle is the dihedral angle of a triangular prism between the base and its lateral face, $\pi /2=90^{\circ }$
• The dihedral angle of an equilateral square pyramid between a triangular face and its base is $\arctan \left(\sqrt{2}\right)\approx 54.7^{\circ }$. Therefore, the dihedral angle of an augmented triangular prism between a square (the lateral face of the triangular prism) and triangle (the lateral face of the equilateral square pyramid) on the edge where the equilateral square pyramid is attached to the square face of the triangular prism, and between two adjacent triangles (the lateral face of both equilateral square pyramids) on the edge where two equilateral square pyramids are attached adjacently to the triangular prism, are $$\begin{array}{cc}\arctan \left(\sqrt{2}\right)+\frac{\pi }{3}& \approx 114.7^{\circ },\\ 2\arctan \left(\sqrt{2}\right)+\frac{\pi }{3}& \approx 169.4^{\circ }.\end{array}$$

Application

In the geometry of chemical compounds, a polyhedron may commonly be visualized an atom cluster surrounding a central atom. The capped trigonal prismatic molecular geometry describes clusters for which this polyhedron is an augmented triangular prism. An example of such compound is the potassium heptafluorotantalate.