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Triangular prism

A triangular prism is a three-dimensional polyhedron with two congruent and parallel triangular bases connected by three parallelogram lateral faces.^[1] It is classified as a pentahedron due to its five faces.^[2] There are two main types: a right triangular prism, where the lateral faces are rectangles perpendicular to the bases, and an oblique triangular prism, where the lateral faces are parallelograms not perpendicular to the bases.^[3] Triangular prisms are notable for their space-filling properties when regular and right, allowing tessellation of three-dimensional space.^[2]

Definition and Classification

Basic Definition

A triangular prism is a polyhedron consisting of two parallel and congruent triangular bases connected by three lateral faces, each of which is a parallelogram.^[4]^[5] This structure distinguishes it from other three-dimensional figures such as pyramids, which feature a single polygonal base with triangular faces converging to an apex rather than two parallel bases, and cylinders, which have curved lateral surfaces instead of flat polygonal faces.^[6] One way to conceptualize the formation of a triangular prism is through the extrusion process, where a triangular polygon is translated along a straight-line axis to generate the solid.^[1] When the translation direction is perpendicular to the plane of the triangle, the resulting figure is a right triangular prism with rectangular lateral faces; oblique variants occur when the translation is at an angle to the base plane.^[7]

Types of Triangular Prisms

Triangular prisms are classified primarily based on the orientation of their lateral edges relative to the bases and the shape of the triangular bases themselves. In a right triangular prism, the lateral edges are perpendicular to the two parallel triangular bases, resulting in lateral faces that are rectangles. This configuration ensures that the prism has a uniform height equal to the length of the lateral edges.^[8]^[1] In contrast, an oblique triangular prism features lateral edges that are not perpendicular to the bases, causing the prism to appear slanted. Here, the lateral faces form parallelograms rather than rectangles, while the bases remain parallel triangles. The height of such a prism is measured as the perpendicular distance between the bases, distinct from the length of the slanted lateral edges.^[8]^[9] A regular triangular prism, also known as a uniform triangular prism, has equilateral triangular bases and square lateral faces, making it a uniform polyhedron where all faces are regular polygons and all edges are of equal length. This specific type is both semiregular and space-filling, allowing tessellations in three-dimensional space.^[2]^[10]^[1] Non-regular variants of triangular prisms arise when the bases are not equilateral, such as isosceles or scalene triangles, paired with rectangular or parallelogram lateral faces depending on whether the prism is right or oblique. For instance, an isosceles triangular base with rectangular sides forms a non-regular right prism, while a scalene base in an oblique configuration yields parallelogram sides. These variations maintain the prismatic structure but lack the symmetry of the regular form.^[11]^[12] Although non-convex triangular prisms are theoretically possible through deformations or non-planar faces, standard classifications and applications focus on convex forms, where the line segment between any two points within the prism lies entirely inside it. Convex triangular prisms, whether right, oblique, regular, or non-regular, form the basis for most geometric studies and practical uses.^[2]^[13]

Geometric Properties

Faces, Edges, and Vertices

A triangular prism is a polyhedron with five faces, consisting of two parallel triangular bases and three quadrilateral lateral faces that connect the corresponding sides of the bases.^[2] The bases are congruent equilateral triangles in the uniform case, while the lateral faces are rectangles for a right prism or parallelograms for an oblique prism.^[14] The structure includes nine edges: six forming the outlines of the two triangular bases (three edges per base) and three additional edges that link the corresponding vertices between the bases. There are six vertices in total, with three vertices on each triangular base, where the lateral edges establish the connections between them.^[15] Regarding connectivity, the graph of the triangular prism is 3-regular, meaning each of the six vertices has degree three—typically incident to two edges from the triangular base and one lateral edge.^[15] Each edge is shared by precisely two faces, ensuring the polyhedron's closed surface topology. For the uniform triangular prism, the Schläfli symbol is denoted as {3} × {}, reflecting its prismatic construction from a triangular base.^[16]

Surface Area and Volume

The volume V of a triangular prism is given by the formula V = area of the triangular base \times height of the prism, where the height is the perpendicular distance between the two parallel bases./10%3A__Geometry/10.07%3A_Volume_and_Surface_Area) For a prism with a regular triangular base, this can equivalently be expressed as V = \frac{1}{2} \times perimeter of the base \times apothem of the base \times height, since the base area is \frac{1}{2} \times perimeter \times apothem./10%3A_Solid_Geometry/10.02%3A_Prisms) The lateral surface area of a triangular prism is the product of the perimeter of the triangular base and the height of the prism.^[17] The total surface area is then twice the area of the triangular base plus the lateral surface area./10%3A__Geometry/10.07%3A_Volume_and_Surface_Area) For a right triangular prism with an equilateral triangular base of side length a and prism height h, the area of each base is \frac{\sqrt{3}}{4} a^{2}.^[18] The volume is therefore V = \frac{\sqrt{3}}{4} a^{2} h. The total surface area is \frac{\sqrt{3}}{2} a^{2} + 3 a h, derived from two bases each contributing \frac{\sqrt{3}}{4} a^{2} and the lateral area $3 a h.^[2] For oblique triangular prisms, the formulas for both volume and surface area use the perpendicular height between the bases rather than the length of the lateral edges.^[19] This ensures the volume accounts for the actual space enclosed, and the lateral surface area reflects the areas of the parallelogram faces correctly as perimeter of base \times perpendicular height.^[17]

Dihedral Angles and Symmetry

In a right triangular prism, the dihedral angles between the triangular bases and the adjacent rectangular lateral faces are all 90°, since the lateral faces are perpendicular to the bases. This right angle configuration arises from the definition of a right prism, where the lateral edges are orthogonal to the base planes.^[20] The dihedral angles between adjacent lateral faces depend on the interior angles of the triangular bases. For a right prism with equilateral triangular bases, these dihedral angles measure 60° along each lateral edge, reflecting the 60° interior angles of the equilateral triangle. In general, for a right triangular prism, the dihedral angle between two adjacent lateral faces along a lateral edge above a base vertex equals the interior angle γ at that base vertex. To calculate this dihedral angle, one can determine the angle between the outward normals to the two lateral faces, which lie in the base plane and are perpendicular to the corresponding base edges; the internal dihedral angle θ satisfies θ = 180° minus the angle between the outward normals, yielding θ = γ for the right prism case.^[20]^[21] The symmetry group of a right regular triangular prism (with equilateral bases and square lateral faces) is the dihedral group D_{3h} of order 12. This group encompasses both rotational and reflectional symmetries, preserving the prism's structure. The rotational subgroup is D_3 of order 6, consisting of the identity, rotations by 120° and 240° about the principal axis passing through the centers of the two bases, and three 180° rotations about axes perpendicular to the principal axis and passing through the midpoints of pairs of opposite lateral edges.^[22]^[23] The full D_{3h} group includes six additional improper isometries: three reflections across vertical planes that contain the principal axis and bisect the bases (each passing through a vertex of one base and the midpoint of the opposite side of the other base), and three rotary reflections (or equivalently, reflections combined with 180° rotations) involving a horizontal mirror plane perpendicular to the principal axis and bisecting the prism's height. These symmetries highlight the prism's threefold rotational symmetry combined with mirror planes, distinguishing it from prisms with different base polygons.^[22]^[24]

Constructions and Relations

Dual Polyhedron

The dual polyhedron of the triangular prism is the triangular bipyramid (also known as the triangular dipyramid), a convex polyhedron consisting of six triangular faces, nine edges, and five vertices.^[25]^[26] This structure is one of the 92 Johnson solids, specifically J_{12}.^[27] For the uniform case, it is classified as a deltahedron due to its composition of equilateral triangular faces.^[25] The dual relationship arises because dipyramids are the reciprocals of prisms in general, with the triangular case preserving the combinatorial structure where the original prism's six vertices correspond to the six faces of the bipyramid, its five faces to the five vertices, and its nine edges to the nine edges.^[25] In terms of geometric correspondence, the two triangular bases of the prism map to the two apical vertices of the bipyramid, while the three lateral (rectangular) faces map to the three vertices forming the equatorial triangle.^[25] The six vertices of the prism, each incident to one triangular base and two lateral faces, become the six triangular faces of the bipyramid, with three faces meeting at each apex (corresponding to one base's vertices) and the remaining faces connecting to the equator. For the uniform triangular prism, where the lateral faces are squares, the dual is a regular-faced triangular bipyramid exhibiting D_{3h} point group symmetry, identical to that of the original prism, which includes rotations and reflections preserving the three-fold axis and horizontal mirror plane.^[25]^[28] The triangular bipyramid is constructed as the polar reciprocal of the triangular prism with respect to a sphere centered at its centroid, a process where each face of the prism is replaced by a vertex at the center of its polar plane, and vice versa, ensuring incidence preservation between vertices, edges, and faces.^[29] This duality is confirmed by the vertex figure of the bipyramid, which is triangular at every vertex—each apex connects to three equatorial vertices forming a triangular section, and each equatorial vertex links to two adjacent equatorial vertices and both apices, also yielding a triangle—mirroring the triangular vertex figures of the original prism and its mixed face types in the uniform case.^[25]

In Polyhedral Compounds

Triangular prisms form polyhedral compounds with other instances of themselves or related polyhedra, often in uniform configurations where the components share a common center and exhibit high symmetry. In uniform polyhedron compounds, triangular prisms appear in several vertex-transitive arrangements where multiple copies interlock without face overlap, maintaining equal edge lengths across components. Examples include the compound of four triangular prisms with chiral octahedral symmetry (order 24), comprising 20 faces (8 triangles and 12 squares), 36 edges, and 24 vertices; the compound of eight triangular prisms, also octahedral but achiral, with 40 faces, 72 edges, and 48 vertices; the chiral compound of ten triangular prisms with icosahedral rotational symmetry (order 60), featuring 50 faces, 90 edges, and 60 vertices; and the achiral compound of twenty triangular prisms, likewise icosahedral, with 100 faces, 180 edges, and 120 vertices.^[30]^[31] These icosahedral compounds possess the rotational symmetry of the regular dodecahedron and serve as components in constructions related to stellated dodecahedra, where triangular prisms contribute to the star polyhedra's edge frameworks.^[32] Triangular prisms also participate in uniform compounds with antiprisms, extending the Archimedean series of prisms and antiprisms into interlocked forms. All such uniform compounds are vertex-transitive, ensuring every vertex environment is identical, and the components interlock seamlessly without volumetric overlap.^[31] These polyhedral compounds were identified through systematic studies of uniform polyhedra initiated by H. S. M. Coxeter and collaborators, whose enumeration of symmetric arrangements laid the groundwork for later complete listings.^[33]

Higher-Dimensional Analogues

In Honeycombs

The triangular prismatic honeycomb is a uniform space-filling tessellation of three-dimensional Euclidean space composed entirely of regular triangular prisms as cells, denoted as the product of the triangular tiling and a line, or {3,3} × {} in Schläfli notation. In this arrangement, twelve regular triangular prisms meet at each vertex, with the structure arising as the Cartesian product of a regular triangular tiling in one plane and an infinite line segment in the perpendicular direction, ensuring complete coverage without gaps or overlaps.^[34] This honeycomb exemplifies efficient packing, achieving a space-filling density of 1 by design, as the equilateral triangular bases tile the plane seamlessly while the rectangular lateral faces align to form continuous columns. It relates to Voronoi diagrams of layered triangular lattices, where the dual structure corresponds to hexagonal prismatic cells bounding regions equidistant from lattice points arranged in triangular patterns across parallel planes. Non-uniform variants incorporate oblique triangular prisms within broader sets of parallelohedra, allowing flexible space-filling arrangements that deviate from right prisms while maintaining translational symmetry across the tiling. These oblique forms adjust the lateral edges to non-perpendicular angles, enabling compatibility with other polyhedra in composite honeycombs that still fill space completely.^[35] In crystallography, triangular prisms serve as models for certain layered crystal structures, such as those in hard-particle systems where stacked two-dimensional layers of triangular prisms form honeycomb-like phases with distinct nucleation pathways and phase transitions. For instance, simulations of hard triangular prisms reveal stable crystal phases characterized by oriented stacking, mimicking behaviors observed in colloidal and molecular crystals.^[36] The triangular prism extends to four-dimensional geometry through prismatic constructions, where it serves as the base polyhedron in the Cartesian product with a line segment, yielding the uniform triangular prismatic 4-polytope. This polychoron consists of seven cells: four triangular prisms and three cubes, derived from the two base triangular prisms and the lateral prisms over the base's faces (two additional triangular prisms from the triangular faces and three cubes from the square faces). Its vertex figure is a rectangular pyramid, and it belongs to the infinite family of prismatic uniform 4-polytopes classified by their disconnected Coxeter diagrams.^[37] A closely analogous 4-polytope is the 3-4 duoprism, constructed as the Cartesian product of a triangle and a square, which features exactly four triangular prisms and three cubes as cells, mirroring the cell composition of the triangular prismatic 4-polytope while emphasizing the prismatic nature in a different product orientation. The tetrahedral prism provides another derivation, incorporating four triangular prisms alongside two regular tetrahedra as cells, with Schläfli symbol {3,3} × {4} in product notation. The dual of the tetrahedral prism is a 4-polytope whose cells are self-dual tetrahedra and triangular bipyramids (the dual of the triangular prism), illustrating faceting relations where the triangular prism's dual influences higher-dimensional vertex figures.^[37] Schläfli symbols generalize the triangular prism's construction from its 3D form {3} × {4}—representing the product of a triangle and a square—to 4D analogues like {3} × {4} × {} for prismatic extensions or the duoprism {3,4}, encapsulating the recursive product structure across dimensions. In the enumeration of the 40 non-prismatic uniform 4-polytopes (beyond the six regular ones), plus the infinite prismatic family, the triangular prism appears as a cell in constructions such as the aforementioned duoprism and tetrahedral prism, and as a ridge figure (the 3D element transverse to ridges) in select uniform polychora with triangular symmetry.^[37] The 4D analogue of the triangular prismatic honeycomb is the product of the 2D triangular tiling with a 2D apeirogon, or {3,3} × { } × { }, which tessellates Euclidean 4-space using triangular prismatic 4-polytopes as cells.^[34]

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