Wikipedia / Augmented triangular prism
Wikipedia / Augmented triangular prism
In geometry, the augmented triangular prism is a polyhedron constructed by attaching an equilateral square pyramid onto one of the square faces of a triangular prism. As a result, it is an example of a Johnson solid. It appears in chemistry as the capped trigonal prismatic molecular geometry.
Construction
The augmented triangular prism is a polyhedron that 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 squaress as its faces. A convex polyhedron whose faces are all regular polygons, but which is not uniform, is a Johnson solid, named after American mathematician Norman W. Johnson.
Related to the aforementioned construction, the augmented triangular prism is a composite polyhedron. A polyhedron is said to be composite if, when being sliced by a plane, the pieces result in two or more convex polyhedra with regular polygonal faces. The augmented triangular prism can be sliced into one equilateral square pyramid and a triangular prism; these are non-composite or elementary polyhedra.
Properties
The surface area is the edge length. All augmented triangular prism edges are equal in length. The augmented triangular prism's surface area is then:The volume, respectively. Therefore, the volume of an augmented triangular prism is:
An augmented triangular prism has three-dimensional symmetry group of order four. That is, there are four rigid transformations that map the shape onto itself. These are:
The do-nothing transformation;
Rotation of 180°, about a line through the pyramid's apex and the midpoint of the opposite side;
Reflection through the plane that contains that line, and which is parallel to the prism's triangular faces; and
The composition of the rotation and reflection.
The dihedral angles of an augmented triangular prism (i.e., the angle between two polygonal faces) can be calculated by examining the prism and the pyramid separately, and adding when appropriate:
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The dihedral angle two adjacent squares is the interior angle.
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Application
In the geometry of chemical compounds, a polyhedron may commonly be visualized as 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.