The elongated triangular bipyramid, elongated triangular dipyramid, or triakis triangular prism is a polyhedron constructed from a triangular prism by attaching two tetrahedra to its bases. It is one of the Johnson solids.
Construction
The elongated triangular bipyramid is constructed from a triangular prism by attaching two regular tetrahedra to its bases, a process known as the elongation. These tetrahedra cover the triangular faces so that the resulting polyhedron has nine faces (six of them are equilateral triangles and three of them are squares), fifteen edges, and eight vertices. A convex polyhedron in which all of the faces are regular polygons is a Johnson solid. The elongated bipyramid is one of them, enumerated as the fourteenth Johnson solid <math> J_{14} </math>.
Properties
thumb|3D model of an elongated triangular bipyramid
The surface area of an elongated triangular bipyramid <math> A </math> is the sum of all polygonal faces' area: six equilateral triangles and three squares. The volume of an elongated triangular bipyramid <math> V </math> can be ascertained by slicing it off into two tetrahedra and a regular triangular prism and then adding their volume. The height of an elongated triangular bipyramid <math> h </math> is the sum of two tetrahedra and a regular triangular prism's height. Therefore, given the edge length <math> a </math>, its height, surface area, and volume are formulated as:
<math display="block"> \begin{align}
h &= \left(\frac{2\sqrt{6{3} + 1 \right)a \approx 2.633a, \\
A &= \left(\frac{3\sqrt{3{2} + 3\right)a^2 \approx 5.598a^2, \\
V &= \left(\frac{\sqrt{2{6} + \frac{\sqrt{3{2} \right) a^3 \approx 0.669a^3.
\end{align}
</math>
It has the same three-dimensional symmetry group as the triangular prism, the dihedral group <math> D_{3 \mathrm{h </math> of order twelve. The dihedral angle of an elongated triangular bipyramid can be calculated by adding the angle of the tetrahedron and the triangular prism:
- the dihedral angle of a tetrahedron between two adjacent triangular faces is <math display="inline"> \arccos \left(\frac{1}{3}\right) \approx 70.5^\circ </math>;
- the dihedral angle of the triangular prism between the square to its bases is <math display="inline"> \frac{\pi}{2} = 90^\circ </math>, and the dihedral angle between square-to-triangle, on the edge where tetrahedron and triangular prism are attached, is <math display="inline"> \arccos \left(\frac{1}{3}\right) + \frac{\pi}{2} \approx 160.5^\circ </math>;
- the dihedral angle of the triangular prism between two adjacent square faces is the internal angle of an equilateral triangle <math display="inline"> \frac{\pi}{3} = 60^\circ </math>.
Appearances
The nirrosula, an African musical instrument woven out of strips of plant leaves, is made in the form of a series of elongated bipyramids with non-equilateral triangles as the faces of their end caps.