Editing Triangular bipyramid (section)

== Special cases ==
=== As a right bipyramid ===
Like other [[bipyramid]]s, a triangular bipyramid can be constructed by attaching two tetrahedra face-to-face.{{r|rajwade}} These tetrahedra cover their triangular base, and the resulting polyhedron has six triangles, five [[Vertex (geometry)|vertices]], and nine edges.{{r|king}} A triangular bipyramid is said to be ''right'' if the tetrahedra are symmetrically regular and both of their [[Apex (geometry)|apices]] are on a line passing through the center of the base; otherwise, it is ''oblique''.{{r|niu-xu|alexandrov}}

[[File:Graph of triangular bipyramid.svg|Graph of a triangular bipyramid|alt=A line drawing with multicolored dots|thumb|left|upright]]
According to [[Steinitz's theorem]], a [[Graph (discrete mathematics)|graph]] can be represented as the [[n-skeleton|skeleton]] of a polyhedron if it is a [[Planar graph|planar]] (the edges of the graph do not cross, but intersect at the point) and [[k-vertex-connected graph|three-connected graph]] (one of any two vertices leaves a connected subgraph when removed). A triangular bipyramid is represented by a graph with nine edges, constructed by adding one vertex to the vertices of a [[wheel graph]] representing [[tetrahedra]].{{r|tutte|ssp}}

Like other right bipyramids, a triangular bipyramid has [[Point groups in three dimensions|three-dimensional point-group symmetry]], the [[dihedral group]] <math> D_{3 \mathrm{h}} </math> of order twelve: the appearance of a triangular bipyramid is unchanged as it rotated by one-, two-thirds, and full angle around the [[Axial symmetry|axis of symmetry]] (a line passing through two vertices and the base's center vertically), and it has [[mirror symmetry]] with any bisector of the base; it is also symmetrical by reflection across a horizontal plane.{{r|ak}} A triangular bipyramid is [[face-transitive]] (or isohedral).{{r|mclean}}
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=== As a Johnson solid ===
{{multiple image
 | image1 = Triangular dipyramid.png
 | alt1 = A triangular bipyramid with regular faces
 | image2 = Triangular bipyramid (symmetric net).svg
 | alt2 = Multicolor, flat image of a triangular bipyramid
 | footer = Triangular bipyramid with regular faces alongside its [[Net (polyhedron)|net]]
 | total_width = 400
}}
[[File:J12 triangular bipyramid.stl|thumb|alt=A grayscale image|3D model of a triangular bipyramid as a Johnson solid]]
If the tetrahedra are regular, all edges of a triangular bipyramid are equal in length and form [[Equilateral triangle|equilateral triangular]] faces. A polyhedron with only equilateral triangles as faces is called a [[deltahedron]]. There are eight convex deltahedra, one of which is a triangular bipyramid with [[regular polygon]]al faces.{{r|trigg}} A convex polyhedron in which all of its faces are regular polygons is the [[Johnson solid]], and every convex deltahedron is a Johnson solid. A triangular bipyramid with regular faces is numbered as the twelfth Johnson solid <math> J_{12} </math>.{{r|uehara}} It is an example of a [[composite polyhedron]] because it is constructed by attaching two [[Tetrahedron|regular tetrahedra]].{{r|timofeenko-2009|berman}}

A triangular bipyramid's surface area <math> A </math> is six times that of each triangle. Its volume <math> V </math> can be calculated by slicing it into two tetrahedra and adding their volume. In the case of edge length <math> a </math>, this is:{{r|berman}}
<math display="block"> \begin{align}
 A &= \frac{3\sqrt{3}}{2}a^2 &\approx 2.598a^2, \\
 V &= \frac{\sqrt{2}}{6}a^3 &\approx 0.238a^3.
\end{align} </math>

The [[dihedral angle]] of a triangular bipyramid can be obtained by adding the dihedral angle of two regular tetrahedra. The dihedral angle of a triangular bipyramid between adjacent triangular faces is that of the regular tetrahedron: 70.5 degrees. In an edge where two tetrahedra are attached, the dihedral angle of adjacent triangles is twice that: 141.1 degrees.{{r|johnson}}
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