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 J_{49}.

Properties

An augmented triangular prism with edge length a has a surface area, calculated by adding six equilateral triangles and two squares' area: Its volume can be obtained by slicing it into a regular triangular prism and an equilateral square pyramid, and adding their volume subsequently: It has three-dimensional symmetry group of the cyclic group 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:

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.