Pentagonal prism
A pentagonal prism is a three-dimensional polyhedron consisting of two parallel pentagonal bases connected by five rectangular lateral faces, forming a heptahedron with seven faces, fifteen edges, and ten vertices.[1] In its regular right form, the bases are regular pentagons and the lateral faces are rectangles of equal width, making it a uniform polyhedron known as U_{76}.[1] Pentagonal prisms appear in various applications, including architecture for faceted structures.[2] The dual polyhedron of the regular pentagonal prism is the pentagonal dipyramid, which connects to broader studies in uniform polyhedra and Archimedean solids.[1]Definition and Construction
Definition
A pentagonal prism is a type of polyhedron consisting of two parallel regular pentagonal bases connected by five rectangular lateral faces.[3] This structure forms a three-dimensional solid where the bases are congruent and oriented identically, with the lateral faces serving as the connecting surfaces between corresponding edges of the pentagons. Pentagonal prisms are distinguished by their orientation: in a right pentagonal prism, the lateral faces are perpendicular to the bases, resulting in lateral edges that are straight and normal to the plane of the bases; in contrast, oblique variants feature lateral edges that are not perpendicular, causing the lateral faces to be parallelograms rather than rectangles.[3] The bases are regular pentagons, with equal side lengths and interior angles of 108 degrees. For the uniform pentagonal prism, a specific case among uniform polyhedra, the lateral faces are squares, with all edges of equal length and regular polygonal faces meeting identically at each vertex.[4] The concept of prisms, including pentagonal forms, originated in ancient Greek geometry as part of studies on solid figures, with Euclid providing the foundational definition in his Elements, Book XI: "A prism is a solid figure contained by planes two of which, namely those which are opposite, are equal, similar, and parallel, while the rest are parallelograms."[5] This early treatment laid the groundwork for understanding prismatic solids. The modern classification of uniform prisms, including the pentagonal variant, was further developed in 20th-century enumerations of uniform polyhedra, building on earlier discoveries of semiregular solids.[4] Visually, a pentagonal prism resembles the extrusion of a regular pentagon along a linear path, creating a cylindrical-like form with flat, multifaceted sides that maintain the pentagonal cross-section throughout its height.[3]Net and Construction Methods
A pentagonal prism can be represented in two dimensions through its net, which unfolds the three-dimensional shape into a flat pattern consisting of two congruent regular pentagons connected by five rectangles, one along each side of the pentagons. These rectangles correspond to the lateral faces, with their widths matching the side lengths of the pentagons and their heights determining the prism's overall height. In the uniform case, where the height equals the side length of the base, the rectangles are squares, resulting in a regular right pentagonal prism. To fold the net into the prism, crease along the edges of the rectangles to raise them perpendicularly from one pentagon base, then align and attach the second pentagon to the free edges of the rectangles, ensuring the faces meet without overlaps or gaps.[1][6][7] Multiple variations of the net exist for a pentagonal prism, arising from different arrangements of the rectangular faces relative to the pentagonal bases while maintaining connectivity without overlapping when folded. A standard unfolding positions the two pentagons at opposite ends, with the five rectangles aligned in a linear strip between them, facilitating easy visualization and assembly. Other configurations may arrange some rectangles adjacent to the bases in a more compact layout, such as fanning them around one pentagon with the second pentagon attached to an outer rectangle, though all valid nets preserve the topology of the seven faces.[8][7] Construction of a pentagonal prism begins with creating the regular pentagonal bases using a compass and straightedge: start by drawing a circle, marking a diameter, constructing a perpendicular bisector, and using successive arcs to locate the five vertices based on the golden ratio proportions inherent to the regular pentagon. The prism is then formed by duplicating the pentagon parallel to itself at a specified height and connecting corresponding vertices with straight lines to form the rectangular lateral faces, effectively extruding the base along the vertical axis. For physical models, print the net on cardstock, cut along the outer edges, score the fold lines, and assemble using tape or glue on the overlapping tabs to secure the seams. In 3D printing, import or create the extruded model in software, then print layer by layer to produce a solid version without manual folding.[9][10][11] Software tools facilitate the generation and manipulation of pentagonal prism nets and models for educational or design purposes. GeoGebra allows users to draw and interact with the net dynamically, adjusting parameters like height to visualize folding in real time. Similarly, Mathematica provides functions to render the net and simulate the 3D assembly, enabling precise control over dimensions and orientations.[12][1]Geometric Elements
Faces, Edges, and Vertices
A pentagonal prism consists of 7 faces: two parallel regular pentagonal bases and five rectangular lateral faces connecting corresponding sides of the bases.[13][14] It has 15 edges in total, comprising 10 edges from the two pentagonal bases (5 per base) and 5 lateral edges that join the corresponding vertices of the bases.[13][14] The prism features 10 vertices, with 5 vertices on each pentagonal base.[13][14] The edges are of two types: the base edges, each of length a corresponding to the side length of the regular pentagon, and the lateral edges, each of length h representing the height (or separation) between the two bases.[14][15] At each vertex, the configuration involves three faces meeting: one pentagonal face and two adjacent rectangular faces. In the uniform variant of the pentagonal prism, these rectangular faces are squares.[13] Topologically, the pentagonal prism satisfies the Euler characteristic \chi = V - E + F = 10 - 15 + 7 = 2, verifying its integrity as a convex polyhedron equivalent to a sphere with genus 0.[16][17]Dimensions and Measures
The pentagonal prism is defined by two primary dimensions: the side length a of each regular pentagonal base and the height h separating the parallel bases. These parameters determine all linear measures of the prism. The apothem r of the base pentagon, representing the perpendicular distance from the center to a side, is given by r = \frac{a}{2 \tan 36^\circ} = \frac{a}{2} \cot \frac{\pi}{5}. This value, approximately $0.6882a, plays a key role in radial distances within the base.[18] In a right pentagonal prism, where the lateral faces are perpendicular to the bases, the dihedral angle between a lateral face and an adjacent base is $90^\circ. The dihedral angle between two adjacent lateral faces is $108^\circ, determined by the geometry of the regular pentagonal base and expressible as \cos^{-1} \left( -\frac{\sqrt{5} - 1}{4} \right). These angles reflect the orthogonal construction and the $108^\circ interior angle of the base pentagon.[19] Face diagonals appear on the pentagonal bases and rectangular lateral faces. Each base pentagon has diagonals of length a \phi, where \phi = \frac{1 + \sqrt{5}}{2} \approx 1.618 is the golden ratio. On each lateral rectangle, the diagonals measure \sqrt{a^2 + h^2}. Space diagonals connect non-coplanar vertices across the bases, with length \sqrt{(a \phi)^2 + h^2} for those spanning the base diagonal projection. These connect vertices separated by two steps around the base perimeter.[18] For the uniform pentagonal prism, where all edges equal a (thus h = a), additional radial measures describe the polyhedron's extent from its center. The inradius of the base (apothem) is \rho = \frac{a}{2} \cot \frac{\pi}{5}. The circumradius R, distance from center to a vertex, is R = \frac{\sqrt{5(15 + 2\sqrt{5})}}{10} a \approx 0.9867 a. The midradius, distance from center to the midpoint of an edge, is m = \frac{\sqrt{10(5 + \sqrt{5})}}{10} a \approx 0.8507 a. Note that the uniform case lacks an insphere, as the distance to bases (h/2 = a/2) differs from the base apothem.[18][19]Physical Properties
Volume
The volume of a pentagonal prism, like any prism, is determined by the product of the area of its base and its height, where the height is the perpendicular distance between the two parallel bases.[20] For a regular pentagonal prism with side length a of the pentagonal bases and height h, the base area A_\text{base} is given by \frac{5}{4} a^2 \cot\left(\frac{\pi}{5}\right), yielding the volume formula V = \frac{5}{4} a^2 h \cot\left(\frac{\pi}{5}\right). [21] This formula can be derived using Cavalieri's principle, which states that two solids have equal volumes if every plane parallel to their bases intersects them in cross-sections of equal area; for a prism, the constant cross-sectional area A_\text{base} along the height h thus gives V = A_\text{base} \cdot h, independent of whether the prism is right or oblique, as long as the perpendicular height is used.[22] Alternatively, in a calculus-based approach, the volume is the integral of the cross-sectional area over the height: V = \int_0^h A_\text{base} \, dz = A_\text{base} \cdot h.[23] For a uniform regular pentagonal prism, where the height equals the side length (h = a), the volume simplifies to V = \frac{5}{4} a^3 \cot\left(\frac{\pi}{5}\right). This emphasizes the prismatic nature, where the volume scales directly with the base area and height, distinguishing it from more complex polyhedra.[20]Surface Area
The total surface area of a right regular pentagonal prism, with side length a of the base pentagon and height h, is given by the formulaS = 2 A_{\text{base}} + 5 a h,
where A_{\text{base}} is the area of one regular pentagonal base.[1] The area of the regular pentagonal base is
A_{\text{base}} = \frac{5}{4} a^2 \cot \frac{\pi}{5},
derived from dividing the pentagon into five congruent isosceles triangles and summing their areas using the cotangent of the central angle \frac{2\pi}{5}.[18] Substituting this into the surface area formula yields
S = 2 \left( \frac{5}{4} a^2 \cot \frac{\pi}{5} \right) + 5 a h = \frac{5}{2} a^2 \cot \frac{\pi}{5} + 5 a h.
The first term accounts for the two identical pentagonal bases, while the second term represents the lateral surface area, which is the product of the base perimeter $5a and the height h.[1] In the uniform case, where the height equals the side length (h = a) and the lateral faces are squares, the surface area simplifies to
S = a^2 \left( \frac{5}{2} \cot \frac{\pi}{5} + 5 \right).
For a unit edge length (a = 1, h = 1), this evaluates numerically to approximately 8.441, confirming the formula's consistency with known values for the regular polyhedron.[1] This surface area can be derived by unfolding the prism into its net, which consists of two regular pentagons separated by five rectangles each of dimensions a \times h. The total area is then the sum of the areas of these seven polygons: $2 A_{\text{base}} + 5 (a h).[1]