Right Prisms

A right prism is a three-dimensional polyhedron consisting of two parallel, congruent polygonal bases connected by rectangular lateral faces, where the lateral edges are perpendicular to the planes of the bases. This perpendicularity ensures that the angle between each lateral face and the bases is 90 degrees, distinguishing it from other prism orientations. The height of a right prism is defined as the perpendicular distance between the two bases, which coincides with the length of the lateral edges.

Right prisms are constructed by linearly extruding a polygonal base along a direction perpendicular to its plane for a fixed distance equal to the height hhh. This process creates uniform cross-sections parallel to the base throughout the prism, resulting in lateral faces that are all rectangles. For instance, starting with a triangle as the base and extruding it perpendicularly produces a right triangular prism.

The orthogonal alignment in right prisms offers advantages in geometric measurements, as the right angles allow for straightforward computations of dimensions and properties without requiring adjustments for slant or obliquity. Examples include the right triangular prism, where the bases are triangles and the lateral faces connect corresponding sides rectangularly, and the right pentagonal prism, featuring pentagonal bases with five rectangular lateral faces.

In a Cartesian coordinate system, the bases of a right prism lie in parallel planes—such as the planes z=0z = 0z=0 and z=hz = hz=h—with the lateral edges aligned parallel to the z-axis, facilitating precise modeling and analysis in three-dimensional space.

Oblique Prisms

An oblique prism is a prism in which the lateral edges are parallel to each other but not perpendicular to the bases, resulting in lateral faces that are parallelograms rather than rectangles.[5] This contrasts with right prisms, where the lateral edges are perpendicular to the bases.[5]

An oblique prism can be obtained by applying a shearing transformation to a right prism, where the shear is performed parallel to the bases, thereby preserving the shape of the bases and the perpendicular height between them.[12] Shearing, as a type of affine transformation, maintains parallelism and thus ensures that the lateral edges remain parallel in the resulting oblique form.

The height of an oblique prism is defined as the perpendicular distance between the two parallel bases, which is distinct from the length of the slanted lateral edges.[5] This perpendicular height is crucial for calculations, as it determines the separation regardless of the obliqueness angle.

A prominent example of an oblique prism is the parallelepiped, particularly the oblique rectangular prism, which features rectangular bases and parallelogram lateral faces. Parallelepipeds are commonly used in modeling crystal lattices, where the unit cell is represented as a parallelepiped to describe the periodic arrangement of atoms in three dimensions.[13]

Due to the volume-preserving nature of the shearing transformation, the volume of an oblique prism equals that of its right prism counterpart with identical base area and perpendicular height, underscoring the affine invariance of volume under such operations.[14]

Uniform Prisms

A uniform prism is a convex polyhedron formed by two parallel regular n-gonal bases connected by n square lateral faces, where all edges are of equal length, ensuring that every face is a regular polygon.[15] This configuration implies a right orientation, with the lateral edges perpendicular to the bases.[16]

The defining property of uniformity in these prisms is vertex-transitivity: the symmetry group of the polyhedron acts transitively on its vertices, meaning any vertex can be mapped to any other via an isometry of the figure.[15] As such, uniform prisms belong to the class of uniform polyhedra, which encompasses the 13 Archimedean solids along with infinite families of prisms and antiprisms; however, uniform prisms themselves constitute an infinite family indexed by n ≥ 3.[17]

Representative examples include the uniform triangular prism, composed of two equilateral triangular bases and three intervening squares, and the uniform hexagonal prism, featuring two regular hexagonal bases flanked by six squares.[18] These structures exemplify the prismatic uniform polyhedra, where the bases determine the polygonal regularity while the squares ensure edge uniformity across the entire figure.[16]

Uniform prisms with bases that tile the plane—such as equilateral triangles (n=3), squares (n=4), or regular hexagons (n=6)—can also tile three-dimensional Euclidean space without gaps or overlaps.[19] For n=4, this yields the cubic honeycomb, a regular tessellation consisting entirely of cubes.

Regular Prisms

A regular prism is a prism whose two parallel bases are congruent regular polygons with nnn sides, where n≥3n \geq 3n≥3. The lateral faces consist of nnn parallelograms connecting the corresponding edges of the bases; in a right regular prism, these are rectangles perpendicular to the bases, while in an oblique regular prism, they are non-rectangular parallelograms due to the lateral shift.[20][5]

Unlike a uniform prism, which requires the lateral faces to be squares—achieved when the prism height equals the side length of the base—a regular prism allows arbitrary height, resulting in non-square rectangular lateral faces for right cases or corresponding parallelograms for oblique ones. This distinction means regular prisms are not necessarily edge-transitive or fully uniform unless the height matches the base side length.[21][22]

For example, a regular pentagonal prism features two equilateral pentagonal bases and five rectangular lateral faces if right, or parallelogram faces if oblique, illustrating how the regularity is confined to the bases without extending to the full edge uniformity.[8]

In regular prisms, the faces meet at equal angles at each vertex, yielding congruent vertex figures across all vertices and ensuring consistent local geometry. However, vertices are fully transitive under the symmetry group for right regular prisms, where the group DnhD_{nh}Dnh​ (of order 4n4n4n) acts transitively.[23][22]

Regular prisms form the foundational structures for uniform prisms and play a key role in polyhedral enumeration, particularly in classifying polyhedra with regular polygonal bases and parallelogram sides.

Volume

The volume of a prism is calculated as the product of the area of its base and the perpendicular height between the two parallel bases, given by the formula V=B×hV = B \times hV=B×h, where BBB is the base area and hhh is the perpendicular distance.[24] This formula applies universally to prisms regardless of the shape of the polygonal base, as long as the bases are congruent and parallel.[25]

Cavalieri's principle establishes that the volume remains invariant under shearing transformations, such as those converting a right prism to an oblique one, because cross-sections parallel to the bases maintain equal areas at corresponding heights.[25] Specifically, if two solids share the same height and identical cross-sectional areas for every plane parallel to the bases, their volumes are equal; thus, an oblique prism with the same base area and perpendicular height as a right prism has the same volume.[25] This invariance holds because shearing preserves the perpendicular height and base area, ensuring no change in the integrated cross-sectional contributions to volume.[25]

For a prism with regular polygonal bases, the base area BBB is B=ns24tan⁡(π/n)B = \frac{n s^2}{4 \tan(\pi/n)}B=4tan(π/n)ns2​, where nnn is the number of sides and sss is the side length.[26] Substituting this into the volume formula yields V=ns2h4tan⁡(π/n)V = \frac{n s^2 h}{4 \tan(\pi/n)}V=4tan(π/n)ns2h​.[26] This expression generalizes the volume for regular prisms, with the base area derived from dividing the polygon into nnn congruent isosceles triangles and summing their areas.[26]

Representative examples illustrate this formula. For a regular triangular prism with equilateral bases of side length sss, the base area is B=34s2B = \frac{\sqrt{3}}{4} s^2B=43​​s2, so the volume is V=34s2hV = \frac{\sqrt{3}}{4} s^2 hV=43​​s2h.[27] For a cube, a special case of a square prism where the height equals the side length aaa, the volume simplifies to V=a3V = a^3V=a3.[28]

The formula extends to prisms with irregular polygonal bases, where BBB is simply the area of the arbitrary polygon, multiplied by the perpendicular height hhh, maintaining the core principle of base extrusion along the height direction.[24]

Surface Area

The surface area of a prism consists of the lateral surface area, which covers the sides connecting the two bases, and the total surface area, which includes the areas of the two bases as well. The lateral surface area AlA_lAl​ is calculated as Al=P×hA_l = P \times hAl​=P×h, where PPP is the perimeter of the base polygon and hhh is the perpendicular height, defined as the shortest distance between the two parallel bases.[29] This formula holds for both right and oblique prisms, as the area of each lateral face—a rectangle in right prisms or a parallelogram in oblique prisms—depends on the base edge length multiplied by the perpendicular height of the face, which corresponds to hhh.

In a right prism, the lateral faces are rectangles, so the lateral surface area is the sum of the areas of these rectangles, each given by the length of a base edge times hhh, yielding the overall Al=P×hA_l = P \times hAl​=P×h.[25] For an oblique prism, the lateral faces are parallelograms, but their areas are still computed using the base edge lengths and the perpendicular height hhh, without adjustment for the slant of the lateral edges; the perimeter PPP remains unchanged from the base polygon's perimeter.[30]

The total surface area AAA is then A=2B+AlA = 2B + A_lA=2B+Al​, where BBB is the area of one base.[29] For a rectangular prism with length lll, width www, and height hhh, this simplifies to A=2(lw+lh+wh)A = 2(lw + lh + wh)A=2(lw+lh+wh), accounting for the two rectangular bases and four lateral rectangular faces.[31] In the case of a regular nnn-gonal prism, where the base is a regular polygon with side length sss, the perimeter is P=nsP = n sP=ns and the base area is B=ns24tan⁡(π/n)B = \frac{n s^2}{4 \tan(\pi/n)}B=4tan(π/n)ns2​, so the total surface area becomes A=2[ns24tan⁡(π/n)]+nshA = 2 \left[ \frac{n s^2}{4 \tan(\pi/n)} \right] + n s hA=2[4tan(π/n)ns2​]+nsh.[25]

Symmetry

The symmetry of a prism is determined by the shape of its bases and the orientation of its lateral edges relative to those bases. For a right uniform prism, where the bases are regular n-gons and the lateral edges are perpendicular to the bases, the symmetry group is the dihedral prismatic group DnhD_{nh}Dnh​ of order 4n4n4n. This group combines the rotational and reflectional symmetries of the regular n-gon base (DnD_nDn​) with additional symmetries along the prism's axis, including a horizontal reflection plane perpendicular to the axis and vertical reflection planes containing the axis.[5]

The full group actions of DnhD_{nh}Dnh​ include n-fold rotations around the principal axis passing through the centers of the two bases, n twofold rotation axes perpendicular to the principal axis and lying in the horizontal plane, and 2n reflection planes: n vertical planes bisecting the base angles or edges, and n dihedral planes bisecting the base sides, along with the horizontal plane. In uniform prisms, all vertices are equivalent under this symmetry, resulting in identical vertex figures at each vertex, typically consisting of a regular n-gon flanked by two squares.[16]

Oblique prisms, in which the lateral edges are not perpendicular to the bases, exhibit reduced symmetry due to the shearing distortion, typically lowering the group to DnD_nDn​ or a subgroup thereof, as the horizontal reflection plane is lost while preserving the base's dihedral symmetries if the bases remain regular.[32]

A representative example is the right triangular prism with equilateral triangular bases, which has symmetry group D3hD_{3h}D3h​ of order 12, featuring threefold rotations around the central axis and three vertical reflection planes. In extended structures such as space-filling tilings or prismatic honeycombs, finite prisms can generate infinite prismatic symmetry groups by translation along the lateral direction, forming wallpaper or space groups with unbounded repetitions.[33]

Schlegel Diagrams

A Schlegel diagram is a perspective projection of a polyhedron onto a plane, obtained by selecting one face as the exterior boundary and projecting the remaining vertices and edges from a point near the center of that face but outside the polyhedron.[34] This representation, named after the German mathematician Victor Schlegel who introduced it in 1886, preserves the combinatorial structure of the polyhedron while flattening it into two dimensions without edge crossings.[35] The outer face serves as the frame, enclosing the projections of all other faces, edges, and vertices.[36]

For prisms, the Schlegel diagram typically projects one base polygon as the outer boundary, with the opposite base appearing as a smaller, similar polygon inside it, connected by radial lines representing the lateral edges.[34] The lateral faces, which are quadrilaterals such as rectangles in right prisms or parallelograms in oblique prisms, project as trapezoids or other quadrilaterals spanning between the inner and outer bases.[37] Construction involves positioning the projection point close to the chosen base and mapping vertices onto a plane parallel to it, often by iteratively adjusting inner vertices toward the centers of their neighboring triangles while fixing the boundary.[37] This method ensures that the diagram accurately depicts the prism's two parallel bases and connecting lateral faces without distortion of their adjacency.

The advantages of Schlegel diagrams for prisms lie in their ability to reveal the full internal connectivity and vertex linkages in a single view, making them particularly useful for analyzing uniform prisms where regular polygonal bases highlight symmetry.[36] Unlike three-dimensional models, these diagrams eliminate hidden elements, allowing clear visualization of how the bases interact with the lateral faces.[36] For instance, the Schlegel diagram of a right square prism (cube) shows an outer square enclosing an inner square, with four connecting lines forming the projections of the side faces, demonstrating the polyhedron's octahedral symmetry group.[34] Similarly, for a triangular prism, the diagram features an outer triangle containing an inner triangle linked by three lines, with three trapezoidal regions representing the lateral faces, aiding in the study of its prismatic structure.[34]

Projections and Unfoldings

Orthographic projections represent prisms through parallel projections onto principal planes, such as the horizontal, frontal, and profile planes, to create multiview drawings that preserve true lengths and angles perpendicular to the line of sight.[38] In these projections, a right prism with its axis perpendicular to the horizontal plane yields a top view showing the true shape of the polygonal base, while the front view displays a rectangle formed by the height and one base edge.[39]

Common orthographic views include the front view, which captures the base and height for a right prism, resulting in a rectangular silhouette; the top view, which outlines the base polygon; and the side view, which also appears as a rectangle for right prisms.[38] For oblique prisms, these views distort: the front view shows parallelograms instead of rectangles due to the inclined axis, and the top view presents a sheared base shape.[39] Isometric projections, a type of axonometric view, provide a three-dimensional illusion by projecting at equal angles (typically 120 degrees) to simulate depth while maintaining proportional dimensions.[38]

Unfoldings, or nets, arrange the faces of a prism in a non-overlapping planar pattern that can be folded back into the three-dimensional form, consisting of two identical polygonal bases connected by a strip of lateral faces.[5] For a right prism, the lateral faces are rectangles forming a continuous band; in oblique prisms, these become parallelograms to reflect the slant.[5] A triangular prism net, for example, features two equilateral triangles as bases flanking three rectangles (or parallelograms for oblique cases), with nine distinct configurations possible depending on the arrangement.[40]

These projections and nets are essential in technical drawing for manufacturing and visualization, enabling accurate pattern layouts for fabrication and precise multiview documentation of prism components.[39]

Frustums

A frustum is the portion of a pyramid between two parallel planes cutting all lateral faces, featuring two non-congruent polygonal bases. It is formed by truncating a pyramid parallel to its base, thereby removing a smaller pyramid cap.[41]

The lateral faces of a frustum consist of trapezoids connecting the two bases. The volume VVV of a frustum is calculated using the formula

where hhh is the perpendicular height between the bases, and B1B_1B1​ and B2B_2B2​ are the areas of the two polygonal bases.[42]

A representative example is the frustum of a square pyramid, characterized by trapezoidal lateral faces with non-parallel slanting edges. These shapes find practical application in engineering, such as in the construction of buckets and hoppers for material handling.[43][44]

In distinction from full prisms, the lateral edges of a frustum are non-parallel and converge toward one another.[45]

Truncated Prisms

A truncated prism is a uniform polyhedron obtained by performing a uniform truncation on a uniform prism, where the vertices are cut off until the original edges are reduced to points, resulting in all edges being of equal length and all faces being regular polygons. This process preserves the convexity and uniformity of the original prism while introducing new faces at the truncated vertices.[15][46]

In the structure of a truncated n-prism, the two original n-gonal bases are transformed into regular 2n-gons, as each original edge of the base is replaced by two new edges separated by the truncation cuts. The n original quadrilateral lateral faces (squares in the uniform case) become regular octagons, with each of the four original edges replaced by two, and new edges added from the truncations at the four vertices. Additionally, each of the 2n original vertices, which are 3-valent, is replaced by a new regular triangular face. Thus, the truncated n-prism has 2 regular 2n-gons, n regular octagons, and 2n regular triangles as its faces. The number of vertices is 6n, and the number of edges is 9n, satisfying Euler's formula for polyhedra.[47][48]

Representative examples include the cases for small n. The truncated triangular prism (n=3) consists of two regular hexagons, three regular octagons, and six regular triangles; it has 18 vertices and 27 edges. The truncated square prism (n=4) features two regular octagons from the bases, four regular octagons from the lateral faces (indistinguishable from the base-derived ones in type), and eight regular triangles. The truncated pentagonal prism (n=5) has two regular decagons, five regular octagons, and ten regular triangles. These examples illustrate the progression in complexity while maintaining uniform edge lengths.[47]

The symmetry of a truncated n-prism is the same as that of the original uniform prism, belonging to the dihedral group D_{nh}, which includes rotations around the principal axis by multiples of 360°/n, reflections through planes containing the axis, and horizontal reflections perpendicular to the axis. This prismatic symmetry ensures vertex-transitivity and the regular nature of all faces.[16]

Truncated prisms for n=3,4,5 are among the convex uniform polyhedra with prismatic symmetry, extending the class of Archimedean solids in broader enumerations of semi-regular polyhedra beyond the 13 with full tetrahedral, octahedral, or icosahedral symmetry.[46][49]

Star and Crossed Prisms

A star prism is a type of non-convex uniform polyhedron formed by two parallel regular star polygon bases connected by rectangular lateral faces, typically squares in the uniform case, where the bases have a density greater than 1.[50] These bases are star polygons, denoted in Schläfli symbol as {n/k} with k > 1, such as the pentagram {5/2}, leading to indentations and a star-like appearance in the overall structure.[51] Unlike convex prisms with simple polygonal bases, star prisms are vertex-transitive but exhibit reduced symmetry, belonging to the dihedral group D_{nh}, and their non-convexity arises from the intersecting edges of the base polygons without the full polyhedron surface self-intersecting in the standard construction.[52]

Crossed prisms represent a further non-convex variant where the two star polygon bases are rotated relative to each other—often by an angle like 36° for pentagonal cases—causing the lateral faces to intersect and producing self-intersecting polyhedra with higher density.[53] In the pentagrammic crossed prism, for instance, two {5/2} pentagrams are linked by triangular lateral faces, resulting in a density of 2 and a structure where vertices lie on a hyperboloid surface, enabling the crossing while maintaining geometric regularity.[54] These figures may form compounds or exhibit compound-like symmetries, with properties including self-intersection of faces and potential embedding on ruled surfaces, distinguishing them from non-crossed star prisms by their intersecting lateral elements.[53]

Examples of star prisms include the pentagrammic prism (uniform polyhedron U78), with 10 vertices, 15 edges, and 7 faces (two pentagrams and five squares), and the stellated triangular prism, a variant using a stellated base configuration to achieve non-convexity through extended facets.[55] Crossed variants, such as the octagrammic crossed prism, demonstrate how rotation introduces hyperbolic geometry influences, with vertices on equilateral hyperboloids.[53] Some star prisms relate to the Kepler–Poinsot regular star polyhedra, like the great stellated dodecahedron {5/2, 3}, through shared use of star polygons and uniform vertex figures, though prisms maintain prismatic symmetry rather than full icosahedral.[51]

Higher-Dimensional Prisms

In higher dimensions, an n-prism is defined as the Cartesian product of an (n-1)-dimensional polytope with a line segment, resulting in an n-dimensional polytope where the two parallel (n-1)-dimensional facets correspond to the copies of the base polytope at the endpoints of the segment.[56] This construction generalizes the familiar 3D prism, with the lateral (n-1)-facets being the products of the (n-2)-facets of the base with the line segment.[56]

Uniform n-prisms extend this idea by requiring the two parallel (n-1)-cells to be regular polytopes and the connecting prism cells to be cubic in their lateral structure, ensuring vertex-transitivity and equal edge lengths throughout.[57] Notable examples include the 4-dimensional tesseract, which can be constructed as a prism over a square base extruded through two dimensions (equivalent to the product of a square and two line segments), featuring eight cubic cells.[58] Similarly, the 5-dimensional penteract (or 5-cube) arises as a prism over a cubic base, with ten tesseractic 4-cells.[59]

The n-dimensional volume of an n-prism is given by the product of the (n-1)-dimensional volume of the base polytope and the length hhh of the line segment, V=Vn−1×hV = V_{n-1} \times hV=Vn−1​×h.[56] The symmetry group of an n-prism is the direct product of the symmetry group of the base polytope and the dihedral group D2hD_{2h}D2h​ acting on the extrusion direction, allowing reflections and rotations that preserve the parallel bases.[60]

Infinite families of uniform prismatic polytopes exist in higher dimensions, including prismatic honeycombs that tile Euclidean n-space, such as the product of an (n-1)-dimensional cubic honeycomb with a line segment, forming space-filling tessellations with uniform cells.[61]

Right Prisms

A right prism is a three-dimensional polyhedron consisting of two parallel, congruent polygonal bases connected by rectangular lateral faces, where the lateral edges are perpendicular to the planes of the bases. This perpendicularity ensures that the angle between each lateral face and the bases is 90 degrees, distinguishing it from other prism orientations. The height of a right prism is defined as the perpendicular distance between the two bases, which coincides with the length of the lateral edges.

Right prisms are constructed by linearly extruding a polygonal base along a direction perpendicular to its plane for a fixed distance equal to the height hhh. This process creates uniform cross-sections parallel to the base throughout the prism, resulting in lateral faces that are all rectangles. For instance, starting with a triangle as the base and extruding it perpendicularly produces a right triangular prism.

The orthogonal alignment in right prisms offers advantages in geometric measurements, as the right angles allow for straightforward computations of dimensions and properties without requiring adjustments for slant or obliquity. Examples include the right triangular prism, where the bases are triangles and the lateral faces connect corresponding sides rectangularly, and the right pentagonal prism, featuring pentagonal bases with five rectangular lateral faces.

In a Cartesian coordinate system, the bases of a right prism lie in parallel planes—such as the planes z=0z = 0z=0 and z=hz = hz=h—with the lateral edges aligned parallel to the z-axis, facilitating precise modeling and analysis in three-dimensional space.

Oblique Prisms

An oblique prism is a prism in which the lateral edges are parallel to each other but not perpendicular to the bases, resulting in lateral faces that are parallelograms rather than rectangles.[5] This contrasts with right prisms, where the lateral edges are perpendicular to the bases.[5]

An oblique prism can be obtained by applying a shearing transformation to a right prism, where the shear is performed parallel to the bases, thereby preserving the shape of the bases and the perpendicular height between them.[12] Shearing, as a type of affine transformation, maintains parallelism and thus ensures that the lateral edges remain parallel in the resulting oblique form.

The height of an oblique prism is defined as the perpendicular distance between the two parallel bases, which is distinct from the length of the slanted lateral edges.[5] This perpendicular height is crucial for calculations, as it determines the separation regardless of the obliqueness angle.

A prominent example of an oblique prism is the parallelepiped, particularly the oblique rectangular prism, which features rectangular bases and parallelogram lateral faces. Parallelepipeds are commonly used in modeling crystal lattices, where the unit cell is represented as a parallelepiped to describe the periodic arrangement of atoms in three dimensions.[13]

Due to the volume-preserving nature of the shearing transformation, the volume of an oblique prism equals that of its right prism counterpart with identical base area and perpendicular height, underscoring the affine invariance of volume under such operations.[14]

Uniform Prisms

A uniform prism is a convex polyhedron formed by two parallel regular n-gonal bases connected by n square lateral faces, where all edges are of equal length, ensuring that every face is a regular polygon.[15] This configuration implies a right orientation, with the lateral edges perpendicular to the bases.[16]

The defining property of uniformity in these prisms is vertex-transitivity: the symmetry group of the polyhedron acts transitively on its vertices, meaning any vertex can be mapped to any other via an isometry of the figure.[15] As such, uniform prisms belong to the class of uniform polyhedra, which encompasses the 13 Archimedean solids along with infinite families of prisms and antiprisms; however, uniform prisms themselves constitute an infinite family indexed by n ≥ 3.[17]

Representative examples include the uniform triangular prism, composed of two equilateral triangular bases and three intervening squares, and the uniform hexagonal prism, featuring two regular hexagonal bases flanked by six squares.[18] These structures exemplify the prismatic uniform polyhedra, where the bases determine the polygonal regularity while the squares ensure edge uniformity across the entire figure.[16]

Uniform prisms with bases that tile the plane—such as equilateral triangles (n=3), squares (n=4), or regular hexagons (n=6)—can also tile three-dimensional Euclidean space without gaps or overlaps.[19] For n=4, this yields the cubic honeycomb, a regular tessellation consisting entirely of cubes.

Regular Prisms

A regular prism is a prism whose two parallel bases are congruent regular polygons with nnn sides, where n≥3n \geq 3n≥3. The lateral faces consist of nnn parallelograms connecting the corresponding edges of the bases; in a right regular prism, these are rectangles perpendicular to the bases, while in an oblique regular prism, they are non-rectangular parallelograms due to the lateral shift.[20][5]

Unlike a uniform prism, which requires the lateral faces to be squares—achieved when the prism height equals the side length of the base—a regular prism allows arbitrary height, resulting in non-square rectangular lateral faces for right cases or corresponding parallelograms for oblique ones. This distinction means regular prisms are not necessarily edge-transitive or fully uniform unless the height matches the base side length.[21][22]

For example, a regular pentagonal prism features two equilateral pentagonal bases and five rectangular lateral faces if right, or parallelogram faces if oblique, illustrating how the regularity is confined to the bases without extending to the full edge uniformity.[8]

In regular prisms, the faces meet at equal angles at each vertex, yielding congruent vertex figures across all vertices and ensuring consistent local geometry. However, vertices are fully transitive under the symmetry group for right regular prisms, where the group DnhD_{nh}Dnh​ (of order 4n4n4n) acts transitively.[23][22]

Regular prisms form the foundational structures for uniform prisms and play a key role in polyhedral enumeration, particularly in classifying polyhedra with regular polygonal bases and parallelogram sides.

Volume

The volume of a prism is calculated as the product of the area of its base and the perpendicular height between the two parallel bases, given by the formula V=B×hV = B \times hV=B×h, where BBB is the base area and hhh is the perpendicular distance.[24] This formula applies universally to prisms regardless of the shape of the polygonal base, as long as the bases are congruent and parallel.[25]

Cavalieri's principle establishes that the volume remains invariant under shearing transformations, such as those converting a right prism to an oblique one, because cross-sections parallel to the bases maintain equal areas at corresponding heights.[25] Specifically, if two solids share the same height and identical cross-sectional areas for every plane parallel to the bases, their volumes are equal; thus, an oblique prism with the same base area and perpendicular height as a right prism has the same volume.[25] This invariance holds because shearing preserves the perpendicular height and base area, ensuring no change in the integrated cross-sectional contributions to volume.[25]

For a prism with regular polygonal bases, the base area BBB is B=ns24tan⁡(π/n)B = \frac{n s^2}{4 \tan(\pi/n)}B=4tan(π/n)ns2​, where nnn is the number of sides and sss is the side length.[26] Substituting this into the volume formula yields V=ns2h4tan⁡(π/n)V = \frac{n s^2 h}{4 \tan(\pi/n)}V=4tan(π/n)ns2h​.[26] This expression generalizes the volume for regular prisms, with the base area derived from dividing the polygon into nnn congruent isosceles triangles and summing their areas.[26]

Representative examples illustrate this formula. For a regular triangular prism with equilateral bases of side length sss, the base area is B=34s2B = \frac{\sqrt{3}}{4} s^2B=43​​s2, so the volume is V=34s2hV = \frac{\sqrt{3}}{4} s^2 hV=43​​s2h.[27] For a cube, a special case of a square prism where the height equals the side length aaa, the volume simplifies to V=a3V = a^3V=a3.[28]

The formula extends to prisms with irregular polygonal bases, where BBB is simply the area of the arbitrary polygon, multiplied by the perpendicular height hhh, maintaining the core principle of base extrusion along the height direction.[24]

Surface Area

The surface area of a prism consists of the lateral surface area, which covers the sides connecting the two bases, and the total surface area, which includes the areas of the two bases as well. The lateral surface area AlA_lAl​ is calculated as Al=P×hA_l = P \times hAl​=P×h, where PPP is the perimeter of the base polygon and hhh is the perpendicular height, defined as the shortest distance between the two parallel bases.[29] This formula holds for both right and oblique prisms, as the area of each lateral face—a rectangle in right prisms or a parallelogram in oblique prisms—depends on the base edge length multiplied by the perpendicular height of the face, which corresponds to hhh.

In a right prism, the lateral faces are rectangles, so the lateral surface area is the sum of the areas of these rectangles, each given by the length of a base edge times hhh, yielding the overall Al=P×hA_l = P \times hAl​=P×h.[25] For an oblique prism, the lateral faces are parallelograms, but their areas are still computed using the base edge lengths and the perpendicular height hhh, without adjustment for the slant of the lateral edges; the perimeter PPP remains unchanged from the base polygon's perimeter.[30]

The total surface area AAA is then A=2B+AlA = 2B + A_lA=2B+Al​, where BBB is the area of one base.[29] For a rectangular prism with length lll, width www, and height hhh, this simplifies to A=2(lw+lh+wh)A = 2(lw + lh + wh)A=2(lw+lh+wh), accounting for the two rectangular bases and four lateral rectangular faces.[31] In the case of a regular nnn-gonal prism, where the base is a regular polygon with side length sss, the perimeter is P=nsP = n sP=ns and the base area is B=ns24tan⁡(π/n)B = \frac{n s^2}{4 \tan(\pi/n)}B=4tan(π/n)ns2​, so the total surface area becomes A=2[ns24tan⁡(π/n)]+nshA = 2 \left[ \frac{n s^2}{4 \tan(\pi/n)} \right] + n s hA=2[4tan(π/n)ns2​]+nsh.[25]

Symmetry

The symmetry of a prism is determined by the shape of its bases and the orientation of its lateral edges relative to those bases. For a right uniform prism, where the bases are regular n-gons and the lateral edges are perpendicular to the bases, the symmetry group is the dihedral prismatic group DnhD_{nh}Dnh​ of order 4n4n4n. This group combines the rotational and reflectional symmetries of the regular n-gon base (DnD_nDn​) with additional symmetries along the prism's axis, including a horizontal reflection plane perpendicular to the axis and vertical reflection planes containing the axis.[5]

The full group actions of DnhD_{nh}Dnh​ include n-fold rotations around the principal axis passing through the centers of the two bases, n twofold rotation axes perpendicular to the principal axis and lying in the horizontal plane, and 2n reflection planes: n vertical planes bisecting the base angles or edges, and n dihedral planes bisecting the base sides, along with the horizontal plane. In uniform prisms, all vertices are equivalent under this symmetry, resulting in identical vertex figures at each vertex, typically consisting of a regular n-gon flanked by two squares.[16]

Oblique prisms, in which the lateral edges are not perpendicular to the bases, exhibit reduced symmetry due to the shearing distortion, typically lowering the group to DnD_nDn​ or a subgroup thereof, as the horizontal reflection plane is lost while preserving the base's dihedral symmetries if the bases remain regular.[32]

A representative example is the right triangular prism with equilateral triangular bases, which has symmetry group D3hD_{3h}D3h​ of order 12, featuring threefold rotations around the central axis and three vertical reflection planes. In extended structures such as space-filling tilings or prismatic honeycombs, finite prisms can generate infinite prismatic symmetry groups by translation along the lateral direction, forming wallpaper or space groups with unbounded repetitions.[33]

Schlegel Diagrams

A Schlegel diagram is a perspective projection of a polyhedron onto a plane, obtained by selecting one face as the exterior boundary and projecting the remaining vertices and edges from a point near the center of that face but outside the polyhedron.[34] This representation, named after the German mathematician Victor Schlegel who introduced it in 1886, preserves the combinatorial structure of the polyhedron while flattening it into two dimensions without edge crossings.[35] The outer face serves as the frame, enclosing the projections of all other faces, edges, and vertices.[36]

For prisms, the Schlegel diagram typically projects one base polygon as the outer boundary, with the opposite base appearing as a smaller, similar polygon inside it, connected by radial lines representing the lateral edges.[34] The lateral faces, which are quadrilaterals such as rectangles in right prisms or parallelograms in oblique prisms, project as trapezoids or other quadrilaterals spanning between the inner and outer bases.[37] Construction involves positioning the projection point close to the chosen base and mapping vertices onto a plane parallel to it, often by iteratively adjusting inner vertices toward the centers of their neighboring triangles while fixing the boundary.[37] This method ensures that the diagram accurately depicts the prism's two parallel bases and connecting lateral faces without distortion of their adjacency.

The advantages of Schlegel diagrams for prisms lie in their ability to reveal the full internal connectivity and vertex linkages in a single view, making them particularly useful for analyzing uniform prisms where regular polygonal bases highlight symmetry.[36] Unlike three-dimensional models, these diagrams eliminate hidden elements, allowing clear visualization of how the bases interact with the lateral faces.[36] For instance, the Schlegel diagram of a right square prism (cube) shows an outer square enclosing an inner square, with four connecting lines forming the projections of the side faces, demonstrating the polyhedron's octahedral symmetry group.[34] Similarly, for a triangular prism, the diagram features an outer triangle containing an inner triangle linked by three lines, with three trapezoidal regions representing the lateral faces, aiding in the study of its prismatic structure.[34]

Projections and Unfoldings

Orthographic projections represent prisms through parallel projections onto principal planes, such as the horizontal, frontal, and profile planes, to create multiview drawings that preserve true lengths and angles perpendicular to the line of sight.[38] In these projections, a right prism with its axis perpendicular to the horizontal plane yields a top view showing the true shape of the polygonal base, while the front view displays a rectangle formed by the height and one base edge.[39]

Common orthographic views include the front view, which captures the base and height for a right prism, resulting in a rectangular silhouette; the top view, which outlines the base polygon; and the side view, which also appears as a rectangle for right prisms.[38] For oblique prisms, these views distort: the front view shows parallelograms instead of rectangles due to the inclined axis, and the top view presents a sheared base shape.[39] Isometric projections, a type of axonometric view, provide a three-dimensional illusion by projecting at equal angles (typically 120 degrees) to simulate depth while maintaining proportional dimensions.[38]

Unfoldings, or nets, arrange the faces of a prism in a non-overlapping planar pattern that can be folded back into the three-dimensional form, consisting of two identical polygonal bases connected by a strip of lateral faces.[5] For a right prism, the lateral faces are rectangles forming a continuous band; in oblique prisms, these become parallelograms to reflect the slant.[5] A triangular prism net, for example, features two equilateral triangles as bases flanking three rectangles (or parallelograms for oblique cases), with nine distinct configurations possible depending on the arrangement.[40]

These projections and nets are essential in technical drawing for manufacturing and visualization, enabling accurate pattern layouts for fabrication and precise multiview documentation of prism components.[39]

Frustums

A frustum is the portion of a pyramid between two parallel planes cutting all lateral faces, featuring two non-congruent polygonal bases. It is formed by truncating a pyramid parallel to its base, thereby removing a smaller pyramid cap.[41]

The lateral faces of a frustum consist of trapezoids connecting the two bases. The volume VVV of a frustum is calculated using the formula

where hhh is the perpendicular height between the bases, and B1B_1B1​ and B2B_2B2​ are the areas of the two polygonal bases.[42]

A representative example is the frustum of a square pyramid, characterized by trapezoidal lateral faces with non-parallel slanting edges. These shapes find practical application in engineering, such as in the construction of buckets and hoppers for material handling.[43][44]

In distinction from full prisms, the lateral edges of a frustum are non-parallel and converge toward one another.[45]

Truncated Prisms

A truncated prism is a uniform polyhedron obtained by performing a uniform truncation on a uniform prism, where the vertices are cut off until the original edges are reduced to points, resulting in all edges being of equal length and all faces being regular polygons. This process preserves the convexity and uniformity of the original prism while introducing new faces at the truncated vertices.[15][46]

In the structure of a truncated n-prism, the two original n-gonal bases are transformed into regular 2n-gons, as each original edge of the base is replaced by two new edges separated by the truncation cuts. The n original quadrilateral lateral faces (squares in the uniform case) become regular octagons, with each of the four original edges replaced by two, and new edges added from the truncations at the four vertices. Additionally, each of the 2n original vertices, which are 3-valent, is replaced by a new regular triangular face. Thus, the truncated n-prism has 2 regular 2n-gons, n regular octagons, and 2n regular triangles as its faces. The number of vertices is 6n, and the number of edges is 9n, satisfying Euler's formula for polyhedra.[47][48]

Representative examples include the cases for small n. The truncated triangular prism (n=3) consists of two regular hexagons, three regular octagons, and six regular triangles; it has 18 vertices and 27 edges. The truncated square prism (n=4) features two regular octagons from the bases, four regular octagons from the lateral faces (indistinguishable from the base-derived ones in type), and eight regular triangles. The truncated pentagonal prism (n=5) has two regular decagons, five regular octagons, and ten regular triangles. These examples illustrate the progression in complexity while maintaining uniform edge lengths.[47]

The symmetry of a truncated n-prism is the same as that of the original uniform prism, belonging to the dihedral group D_{nh}, which includes rotations around the principal axis by multiples of 360°/n, reflections through planes containing the axis, and horizontal reflections perpendicular to the axis. This prismatic symmetry ensures vertex-transitivity and the regular nature of all faces.[16]

Truncated prisms for n=3,4,5 are among the convex uniform polyhedra with prismatic symmetry, extending the class of Archimedean solids in broader enumerations of semi-regular polyhedra beyond the 13 with full tetrahedral, octahedral, or icosahedral symmetry.[46][49]

Star and Crossed Prisms

A star prism is a type of non-convex uniform polyhedron formed by two parallel regular star polygon bases connected by rectangular lateral faces, typically squares in the uniform case, where the bases have a density greater than 1.[50] These bases are star polygons, denoted in Schläfli symbol as {n/k} with k > 1, such as the pentagram {5/2}, leading to indentations and a star-like appearance in the overall structure.[51] Unlike convex prisms with simple polygonal bases, star prisms are vertex-transitive but exhibit reduced symmetry, belonging to the dihedral group D_{nh}, and their non-convexity arises from the intersecting edges of the base polygons without the full polyhedron surface self-intersecting in the standard construction.[52]

Crossed prisms represent a further non-convex variant where the two star polygon bases are rotated relative to each other—often by an angle like 36° for pentagonal cases—causing the lateral faces to intersect and producing self-intersecting polyhedra with higher density.[53] In the pentagrammic crossed prism, for instance, two {5/2} pentagrams are linked by triangular lateral faces, resulting in a density of 2 and a structure where vertices lie on a hyperboloid surface, enabling the crossing while maintaining geometric regularity.[54] These figures may form compounds or exhibit compound-like symmetries, with properties including self-intersection of faces and potential embedding on ruled surfaces, distinguishing them from non-crossed star prisms by their intersecting lateral elements.[53]

Examples of star prisms include the pentagrammic prism (uniform polyhedron U78), with 10 vertices, 15 edges, and 7 faces (two pentagrams and five squares), and the stellated triangular prism, a variant using a stellated base configuration to achieve non-convexity through extended facets.[55] Crossed variants, such as the octagrammic crossed prism, demonstrate how rotation introduces hyperbolic geometry influences, with vertices on equilateral hyperboloids.[53] Some star prisms relate to the Kepler–Poinsot regular star polyhedra, like the great stellated dodecahedron {5/2, 3}, through shared use of star polygons and uniform vertex figures, though prisms maintain prismatic symmetry rather than full icosahedral.[51]

Higher-Dimensional Prisms

In higher dimensions, an n-prism is defined as the Cartesian product of an (n-1)-dimensional polytope with a line segment, resulting in an n-dimensional polytope where the two parallel (n-1)-dimensional facets correspond to the copies of the base polytope at the endpoints of the segment.[56] This construction generalizes the familiar 3D prism, with the lateral (n-1)-facets being the products of the (n-2)-facets of the base with the line segment.[56]

Uniform n-prisms extend this idea by requiring the two parallel (n-1)-cells to be regular polytopes and the connecting prism cells to be cubic in their lateral structure, ensuring vertex-transitivity and equal edge lengths throughout.[57] Notable examples include the 4-dimensional tesseract, which can be constructed as a prism over a square base extruded through two dimensions (equivalent to the product of a square and two line segments), featuring eight cubic cells.[58] Similarly, the 5-dimensional penteract (or 5-cube) arises as a prism over a cubic base, with ten tesseractic 4-cells.[59]

The n-dimensional volume of an n-prism is given by the product of the (n-1)-dimensional volume of the base polytope and the length hhh of the line segment, V=Vn−1×hV = V_{n-1} \times hV=Vn−1​×h.[56] The symmetry group of an n-prism is the direct product of the symmetry group of the base polytope and the dihedral group D2hD_{2h}D2h​ acting on the extrusion direction, allowing reflections and rotations that preserve the parallel bases.[60]

Infinite families of uniform prismatic polytopes exist in higher dimensions, including prismatic honeycombs that tile Euclidean n-space, such as the product of an (n-1)-dimensional cubic honeycomb with a line segment, forming space-filling tessellations with uniform cells.[61]