Augmented triangular prism explained

Type:Johnson
Faces:6 triangles
2 squares
Edges:13
Vertices:7
Symmetry:

C2v

Vertex Config:

\begin{align} &2 x (3 x 42)+\\ &1 x (34)+\\ &4 x (33 x 4)\end{align}

Net:Johnson solid 49 net.png

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 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 6 equilateral triangles and 2 squares as its faces. A convex polyhedron in which all faces are regular is Johnson solid, and the augmented triangular prism is among them, enumerated as 49th Johnson solid

J49

.

Properties

An augmented triangular prism with edge length

a

has a surface area, calculated by adding six equilateral triangles and two squares' area: \fraca^2 \approx 4.598a^2. Its volume can be obtained by slicing it into a regular triangular prism and an equilateral square pyramid, and adding their volume subsequently: \fraca^3 \approx 0.669a^3.

It has three-dimensional symmetry group of the cyclic group

C2v

of order 4. Its dihedral angle can be calculated by adding the angle of an equilateral square pyramid and a regular triangular prism in the following:

\pi/3=60\circ

.

\arctan \left(\sqrt\right) + \frac &\approx 114.7^\circ, \\ 2 \arctan \left(\sqrt\right) + \frac &\approx 169.4^\circ.\end

Application

In the geometry of chemical compounds, a polyhedron may commonly visualize 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.

See also