The biaugmented triangular prism is a polyhedron constructed from a triangular prism by attaching two equilateral square pyramids onto two of its square faces. It is an example of a Johnson solid and of a composite polyhedron. It can be found in stereochemistry as a structural chemical compound called the bicapped trigonal prismatic molecular geometry.

Construction

The biaugmented triangular prism is constructed by attaching two equilateral square pyramids to the faces of a triangular prism; this prism is uniform, having the faces shaped as two equilateral triangles and three squares. These pyramids cover two square faces with eight triangles, resulting in a polyhedron with seventeen faces in total: ten equilateral triangles and one square. The process of construction is known as an augmentation, having two pyramids gives the prefix "bi-". In addition, the resulting polyhedron also has seventeen edges and eight vertices. The biaugmented triangular prism is classified as a Johnson solid, after American mathematician Norman W. Johnson, who listed the 92 convex polyhedra with regular polygonal, marking the 50th Johnson solid.

Related to the aforementioned construction, the biaugmented triangular prism is composite. 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 biaugmented triangular prism can be sliced into two equilateral square pyramids and a triangular prism, three of which are non-composite or elementary polyhedra.

Properties

is the edge length. All edges of the biaugmented triangular prism are equal in length. The biaugmented triangular prism's surface area is then:

, respectively. Therefore, the volume of a biaugmanted triangular prism is:

The biaugmented triangular prism has three-dimensional symmetry group of two-fold pyramidal symmetry of order 4. Its dihedral angle (i.e., the angle between two polygonal faces) can be calculated by adding the angle of an equilateral square pyramid and a regular triangular prism in the following:

• . The dihedral angle of a triangular prism between two adjacent square faces is the internal angle

Application

The biaugmented triangular prism can be found in stereochemistry. In this subfield of chemistry which studies molecule structures, the biaugmented triangular prism is a structural shape of the chemical compound known as the bicapped trigonal prismatic molecular geometry. The structure contains eight vertices representing the atoms, commonly found in transition metal other than the chemical structure of square antiprism and the snub disphenoid. Examples of the structure are plutonium tribromide PuBr3 adopted by bromides and iodides of the lanthanides and actinides, and the anionic octaflouride of octafluorozirconate(IV) .

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