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Pentagonal pyramid

A pentagonal pyramid is a three-dimensional polyhedron consisting of a pentagonal base and five triangular lateral faces that converge at a single apex point.^[1] It is classified as a pyramid based on its polygonal base and triangular sides meeting at a common vertex.^[2] This geometric figure has 6 faces (one pentagonal base and five triangles), 10 edges (five forming the base and five connecting the base vertices to the apex), and 6 vertices (five at the base and one at the apex).^[1] In a regular pentagonal pyramid, the base is a regular pentagon, and the apex is positioned directly above the center of the base, resulting in congruent isosceles triangular lateral faces.^[2] The structure satisfies Euler's formula for polyhedra, with the characteristic V - E + F = 2, confirming its topological properties as a convex polyhedron.^[3] The volume V of a pentagonal pyramid is calculated as one-third the product of the base area and the height: V = \frac{1}{3} B h, where B is the area of the pentagonal base and h is the perpendicular height from the base to the apex.^[4] For a regular pentagonal base with side length s and apothem a, the base area B = \frac{5}{2} s a. The total surface area S comprises the base area plus the lateral surface area: S = B + \frac{1}{2} P l, where P is the perimeter of the base (P = 5s) and l is the slant height of the lateral faces.^[1]^[2] These formulas apply particularly to regular forms, enabling precise computations in geometric analysis and applications.

Definition and Classification

Definition

A pyramid is a three-dimensional polyhedron consisting of a polygonal base and triangular faces that connect the base to a single vertex known as the apex, where all the lateral edges converge.^[5] This structure arises by extending the base polygon linearly toward the apex, forming a solid with non-parallel lateral faces.^[6] A pentagonal pyramid is a specific type of pyramid featuring a pentagon as its base and five triangular lateral faces that meet at the apex.^[7] The base serves as the foundational polygon, while the apex acts as the common vertex linking the tips of the base's sides via the lateral edges.^[1] In ideal configurations, the base is a regular pentagon, ensuring uniformity in the base's sides and angles.^[7] Unlike prisms, which feature two parallel polygonal bases connected by rectangular or parallelogram lateral faces, a pentagonal pyramid has only one base and tapers to a single apex, resulting in converging rather than parallel faces.^[8] This distinction highlights the pyramid's role in representing tapered solids in geometry, distinct from the uniform extrusion seen in prisms.^[9]

Classification

Pentagonal pyramids are classified primarily by the regularity of their pentagonal base, the positioning of the apex relative to the base, and their overall polyhedral properties. A general pentagonal pyramid features an arbitrary pentagon as its base, with five triangular lateral faces connecting the base to the apex, allowing for irregular base shapes without specific symmetry requirements.^[10] Based on apex positioning, pentagonal pyramids are further divided into right and oblique variants. In a right pentagonal pyramid, the apex is located directly above the geometric center (centroid) of the base, ensuring the line from the apex to the center is perpendicular to the base plane.^[11] Conversely, an oblique pentagonal pyramid has its apex offset from the center, resulting in slanted lateral edges and faces that are not perpendicular to the base.^[11] A regular pentagonal pyramid possesses a regular pentagon base—characterized by five equal sides and equal interior angles—and congruent isosceles triangular lateral faces, achieving a high degree of uniformity in its structure.^[7] A special case where the lateral faces are equilateral triangles and all edges are of equal length qualifies as Johnson solid J₂, a strictly convex polyhedron with regular faces that is neither Platonic, Archimedean, nor a prism or antiprism.^[10]^[12] While convex pentagonal pyramids form the standard focus of geometric study due to their simplicity and adherence to Euler's polyhedral formula, concave variants exist, such as those incorporating star-shaped or indented pentagonal elements for non-convex configurations.^[13] However, this article emphasizes convex forms. The convex pentagonal pyramid is one of seven topologically distinct convex hexahedra, alongside structures like the triangular dipyramid and cube, distinguished by their unique vertex-edge-face connectivities.^[14]

Geometric Structure

Faces, Edges, and Vertices

A pentagonal pyramid consists of six faces: one pentagonal base and five triangular lateral faces that converge at the apex.^[1]^[15] The base face is a five-sided polygon, while each lateral face is an isosceles or equilateral triangle depending on the pyramid's regularity, connecting the base perimeter to the single apex point.^[16] The structure includes ten edges in total: five forming the pentagonal base and five additional edges extending from each base vertex to the apex.^[1]^[15] These edges define the boundaries where faces meet, with the base edges creating a closed polygonal cycle and the lateral edges radiating outward from the apex. There are six vertices: five located at the corners of the pentagonal base and one at the apex.^[1]^[15] Each base vertex connects to two adjacent base vertices and to the apex, while the apex connects exclusively to the five base vertices. In terms of connectivity, each triangular lateral face shares one edge with the pentagonal base and two edges with adjacent lateral faces, forming a continuous envelope around the base.^[16] The base itself is a simple polygonal cycle of five edges linking the base vertices. From a graph theory perspective, the skeleton of the pentagonal pyramid is a wheel graph W_6, equivalent to a complete bipartite graph K_{1,5} (representing the star from the apex to the base) augmented by the five edges of the pentagonal cycle.^[17]

Symmetry

The regular pentagonal pyramid exhibits the point group symmetry C_{5v}, characterized by a principal five-fold rotation axis and five vertical mirror planes containing that axis.^[18] This symmetry group applies specifically to the case where the base is a regular pentagon and the apex is positioned directly above the base center, ensuring all lateral faces are congruent isosceles triangles.^[19] The C_{5v} group has an order of 10, consisting of five proper rotations and five improper rotations (reflections).^[18] The rotational subgroup is cyclic of order 5, including the identity and rotations by $72^\circ, $144^\circ, $216^\circ, and $288^\circ around the five-fold axis that passes through the apex and the center of the base.^[20] These rotations map the base pentagon onto itself and cycle the lateral faces accordingly, preserving the overall structure.^[18] Complementing the rotations are five vertical mirror planes (\sigma_v), each containing the five-fold axis and bisecting the pyramid by passing through one base vertex and the midpoint of the opposite base edge.^[18] These reflections swap pairs of lateral faces while fixing the plane of symmetry, contributing to the pyramid's reflectional invariance.^[21] The full C_{5v} symmetry ensures that the regular pentagonal pyramid remains unchanged under any of these 10 operations, underscoring its geometric uniformity and facilitating applications in crystallography and molecular modeling where such five-fold symmetry occurs, as in certain polyatomic ions like \ce{IOF5^2-}.^[18] In contrast, a pentagonal pyramid with an irregular base lacks this high symmetry, typically reducing to no rotational or reflectional symmetries beyond the identity due to non-congruent sides and angles.^[21]

Mathematical Properties

Dihedral Angles

In a regular pentagonal pyramid, where the base is a regular pentagon and the five lateral faces are equilateral triangles (corresponding to Johnson solid J2), there are two distinct types of dihedral angles: those between two adjacent lateral faces and those between a lateral face and the base. The dihedral angle between two adjacent triangular faces is \arccos\left(-\frac{\sqrt{5}}{3}\right) \approx 138.19^\circ.^[16] The dihedral angle between a triangular face and the pentagonal base is \arccos\left(\sqrt{\frac{5 + 2\sqrt{5}}{15}}\right) \approx 37.38^\circ.^[16] These angles can be derived by computing the normals to the adjacent faces and finding the angle between those normals. For instance, place the pyramid in Cartesian coordinates with the apex at (0, 0, h) where h = \sqrt{\frac{5 - \sqrt{5}}{10}} and the base vertices in the xy-plane at positions such as \left(\pm \frac{1}{2}, -\sqrt{\frac{5 + 2\sqrt{5}}{20}}, 0\right), \left(\pm \frac{1 + \sqrt{5}}{4}, \sqrt{\frac{5 - \sqrt{5}}{40}}, 0\right), and \left(0, \sqrt{\frac{5 + \sqrt{5}}{10}}, 0\right) (scaled for unit edge length). Vectors in each face are crossed to obtain the face normals \mathbf{n_1} and \mathbf{n_2}; the dihedral angle \theta satisfies \cos \theta = -\frac{\mathbf{n_1} \cdot \mathbf{n_2}}{||\mathbf{n_1}|| \, ||\mathbf{n_2}||}, accounting for the internal angle in the convex polyhedron. Alternatively, spherical trigonometry on the unit sphere centered at a vertex along the common edge projects the face angles, allowing computation via the spherical law of cosines for the angle between great circles representing the faces.^[16]^[22] The C_{5v} symmetry of the regular pentagonal pyramid ensures all triangle-triangle dihedral angles are identical and all triangle-base angles are identical. For oblique or irregular pentagonal pyramids, these dihedral angles vary depending on the apex position and base irregularities, lacking the uniform exact values of the regular case.^[16]

Euler Characteristic

The Euler characteristic \chi of a polyhedron, defined as \chi = V - E + F where V is the number of vertices, E the number of edges, and F the number of faces, provides a topological invariant that verifies its structure. For the pentagonal pyramid, V = 6, E = 10, and F = 6, so \chi = 6 - 10 + 6 = 2.^[17]^[23] This result of \chi = 2 confirms the pentagonal pyramid as a convex polyhedron homeomorphic to a sphere.^[24] For any pyramid with an n-gonal base, V = n + 1, E = 2n, and F = n + 1, yielding \chi = (n + 1) - 2n + (n + 1) = 2; substituting n = 5 reproduces the value above.^[17] The Euler characteristic distinguishes convex polyhedra from non-polyhedral surfaces, as all such polyhedra consistently exhibit \chi = 2, offering a fundamental check of their spherical topology.^[24] The counts of vertices, edges, and faces follow directly from the pyramid's base and apex configuration.^[17]

Formulas and Calculations

Volume

The volume V of any pyramid, including a pentagonal pyramid, is given by the formula V = \frac{1}{3} A_b h, where A_b is the area of the base and h is the perpendicular height from the base to the apex.^[25] For a pentagonal pyramid with a regular pentagonal base of side length s, the base area is A_b = \frac{1}{4} \sqrt{25 + 10 \sqrt{5}} \, s^2, which is equivalent to A_b = \frac{5}{4} s^2 \cot(\pi/5). Substituting yields V = \frac{1}{12} s^2 h \sqrt{25 + 10 \sqrt{5}}.^[10] In the special case of a regular pentagonal pyramid where all edges, including the base sides and lateral edges, have equal length a (corresponding to Johnson solid J_2), the volume simplifies to V = \frac{5 + \sqrt{5}}{24} a^3 \approx 0.30150 a^3.^[10] This formula arises by first determining the height h = \sqrt{a^2 - R^2}, where R = \frac{a}{2 \sin(\pi/5)} is the circumradius of the base pentagon, and then substituting into the general volume expression.^[10] The general volume formula can be derived using Cavalieri's principle, which states that two solids of equal height have the same volume if their cross-sectional areas parallel to the base are equal at every level.^[26] For a pyramid, cross-sections parallel to the base are similar pentagons scaled by the factor (1 - z/h)^2, where z is the distance from the apex; integrating the cross-sectional area from the apex to the base gives V = \frac{1}{3} A_b h.^[25] Alternatively, the same result follows from direct integration of the scaling factor over the height.^[17]

Surface Area

The surface area of a pentagonal pyramid is the sum of the area of its pentagonal base and the areas of its five triangular lateral faces. For a regular pentagonal pyramid, defined as one with a regular pentagonal base of side length a and the apex positioned directly above the center of the base, the total surface area S is given by S = A_b + L, where A_b is the base area and L is the lateral surface area.^[10] The base area A_b of a regular pentagon is A_b = \frac{1}{4} \sqrt{25 + 10 \sqrt{5}} \, a^2 \approx 1.72048 a^2. This formula is derived by dividing the pentagon into three triangles from the center and using trigonometric relations, such as A_b = \frac{5}{4} a^2 \cot(\pi/5).^[27] The lateral surface area L is L = \frac{5}{2} a l, where l is the slant height, the perpendicular distance from the apex to the midpoint of a base edge. The slant height is l = \sqrt{h^2 + r^2}, with h the pyramid height and r the apothem (distance from center to midpoint of a base side), given by r = \frac{a}{2 \tan(36^\circ)} = \frac{a \sqrt{25 + 10 \sqrt{5}}}{10} \approx 0.68819 a. Each lateral face is an isosceles triangle with base a and equal sides equal to the lateral edge length; its area can be found using the formula for the area of a triangle with base a and height l, or alternatively via Heron's formula applied to the side lengths a, e, e (where e = \sqrt{h^2 + R^2} and R \approx 0.85065 a is the circumradius of the base).^[10] In the special case of a regular pentagonal pyramid with all edges equal to a (where the lateral faces are equilateral triangles), the total surface area simplifies to S = \frac{a^2}{2} \sqrt{\frac{5}{2} \left(10 + \sqrt{5} + \sqrt{75 + 30 \sqrt{5}}\right)} \approx 3.88554 a^2. Here, the base area remains \frac{1}{4} \sqrt{25 + 10 \sqrt{5}} \, a^2, while each equilateral triangular face has area \frac{\sqrt{3}}{4} a^2 \approx 0.43301 a^2, so the lateral contribution is $5 \times \frac{\sqrt{3}}{4} a^2 \approx 2.16506 a^2. This configuration corresponds to the Johnson solid J2, and the formula arises from substituting the specific height h = \frac{a \sqrt{10 + 2 \sqrt{5}}}{4} that equalizes all edges.^[10] For an oblique pentagonal pyramid, where the apex is not above the base center, the base area is unchanged, but the lateral faces are general scalene triangles, requiring individual area calculations. The area of each triangular face can be computed exactly using the vector cross product method: for vertices apex A and base points B, C, the area is \frac{1}{2} \| \overrightarrow{AB} \times \overrightarrow{AC} \|. This approach accounts for the non-perpendicular orientation without approximation, summing over all five faces for the lateral total.

Coordinate Representations

Cartesian Coordinates

A regular pentagonal pyramid can be positioned in Cartesian coordinates with its base lying in the xy-plane, centered at the origin, and the apex directly above the center along the positive z-axis. The apex is located at (0, 0, h), where h > 0 is the height of the pyramid. The five base vertices are positioned at (r \cos(2\pi k / 5), r \sin(2\pi k / 5), 0) for k = 0, 1, 2, 3, 4, where r is the circumradius of the regular pentagonal base.^[16]^[28] For a base with side length a, the circumradius is given by r = \frac{a}{2 \sin(\pi / 5)}. To achieve equal edge lengths throughout (as in the Johnson solid J_2), the height h is chosen such that the lateral edges equal a; this requires solving \sqrt{r^2 + h^2} = a, yielding h = \sqrt{a^2 - r^2}. Substituting the expression for r provides the specific value h = a \sqrt{\frac{5 - \sqrt{5}}{10}} \approx 0.5257 a.^[16]^[29] An alternative set of coordinates for the regular pentagonal pyramid (Johnson solid J_2) with all edges of length 2 leverages the golden ratio \phi = \frac{[1](/page/1) + \sqrt{5}}{2} and derives from a subset of the regular icosahedron's vertices. These are: (\pm [1](/page/1), 0, \phi), (0, \pm \phi, [1](/page/1)), and (\phi, \pm [1](/page/1), 0). This embedding orients the pyramid obliquely, with no base parallel to a coordinate plane, but preserves the C_{5v} symmetry group. To normalize for unit edge length a = [1](/page/1), scale all coordinates by $1/2: (\pm 1/2, 0, \phi/2), (0, \pm \phi/2, 1/2), and (\phi/2, \pm 1/2, 0).^[29]^[16] Explicit upright coordinates for the unit-edge regular pentagonal pyramid (base at z=0, apex above) are: base vertices at \left(\pm \frac{1}{2}, -\sqrt{\frac{5 + 2\sqrt{5}}{20}}, 0\right), \left(\pm \frac{1 + \sqrt{5}}{4}, \sqrt{\frac{5 - \sqrt{5}}{40}}, 0\right), \left(0, \sqrt{\frac{5 + \sqrt{5}}{10}}, 0\right); apex at \left(0, 0, \sqrt{\frac{5 - \sqrt{5}}{10}}\right). These positions ensure all edges measure 1 and can be derived by solving distance constraints between vertices.^[16] To place the pyramid in an arbitrary orientation, apply a 3D rotation matrix to the above coordinates. A general rotation is composed of Euler angles or represented by an orthogonal matrix R with \det(R) = 1, transforming vertices \mathbf{v} to R \mathbf{v} + \mathbf{t}, where \mathbf{t} is a translation vector. For example, a rotation around the z-axis by angle \theta uses the matrix

\begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}.

Such transformations maintain the geometric properties while allowing flexible positioning in space.

Placement and Orientation

In the standard placement of a regular pentagonal pyramid in three-dimensional Cartesian space, the base is positioned as a regular pentagon lying in the xy-plane and centered at the origin, with the apex located directly above the center along the positive z-axis to form a right pyramid. This orientation aligns the five-fold rotational symmetry axis with the z-axis, facilitating computations in geometry and computer graphics. The vertex coordinates for such a placement, with edge length normalized to 1, include base points such as (\pm \frac{1}{2}, -\sqrt{\frac{5 + 2\sqrt{5}}{20}}, 0) and (\pm \frac{1 + \sqrt{5}}{4}, \sqrt{\frac{5 - \sqrt{5}}{40}}, 0), alongside the apex at (0, 0, \sqrt{\frac{5 - \sqrt{5}}{10}}).^[16] Orientation variations of the pentagonal pyramid can be achieved through rotations around its principal five-fold axis or reflections across mirror planes, preserving the right pyramidal structure while adjusting the azimuthal alignment of the base. For instance, discrete rotations by multiples of $72^\circ around the z-axis map the base vertices to one another in this standard setup. To obtain an oblique pentagonal pyramid, where the apex is displaced laterally from the perpendicular above the base center, a shear transformation can be applied to the right pyramid's coordinates, shifting the apex parallel to the base plane without altering the base itself.^[17]^[30] The centroid, or center of mass assuming uniform density, of a pentagonal pyramid in its standard placement lies along the axis of symmetry at a distance of h/4 from the base toward the apex, where h is the height; thus, for a base at z=0 and apex at z=h, the centroid coordinates are (0, 0, h/4). This position is derived from the general formula for the centroid of any pyramidal solid, located at one-fourth the height from the base along the line connecting the base centroid to the apex. Knowledge of the centroid and the axis-aligned bounding box—spanning the minimum and maximum x, y, and z coordinates of the vertices—is essential for applications such as collision detection in simulations and efficient rendering in 3D graphics software.^[31] Due to its self-dual property, the dual of a pentagonal pyramid is another pentagonal pyramid, allowing for an orientation where the roles of vertices and faces are interchanged while maintaining a similar axial placement; in this dual configuration, the original apex corresponds to the dual's base center, and the original base faces map to the dual's apical vertex.^[17]^[16]

Applications

In Polyhedra and Geometry

The pentagonal pyramid serves as a fundamental building block in the construction of more complex polyhedra, particularly through augmentation processes where it is attached to the faces of Platonic solids. For instance, attaching a pentagonal pyramid to each of the 12 pentagonal faces of a regular dodecahedron yields the pentakis dodecahedron, a Catalan solid with 60 triangular faces.^[32] Similarly, augmenting a regular dodecahedron with suitably proportioned pentagonal pyramids, extended until their bases coincide with the original faces, produces the small stellated dodecahedron, one of the four Kepler–Poinsot polyhedra.^[33] As the second Johnson solid (J₂), the pentagonal pyramid is one of the 92 strictly convex polyhedra with regular faces but no regular vertex figures, classified by Norman Johnson in 1966.^[34] This classification highlights its role among deltahedra and other faceted forms that bridge Platonic and Archimedean solids. In polyhedral compounds and extensions, the pentagonal pyramid forms a key component of the gyroelongated pentagonal pyramid (Johnson solid J₁₁), constructed by affixing a pentagonal antiprism to its base, resulting in a structure with 16 faces including the original pyramid's elements.^[35] Derivatives involving the bilunabirotunda (J₉₁) also incorporate pentagonal pyramidal motifs in near-miss constructions or pseudopyramids, extending its utility in Johnson solid families.^[36] Unlike the triangular pyramid, which is a regular tetrahedron and thus a simplex in three dimensions, the pentagonal pyramid introduces a non-simplicial base, enabling greater flexibility in polyhedral assemblies and allowing for irregular vertex configurations.^[12] Its volume scales nonlinearly with the number of base sides compared to lower-sided pyramids, providing a basis for understanding volumetric growth in pyramidal extensions of regular polygons.^[19]

In Chemistry and Materials Science

In stereochemistry, pentagonal pyramidal geometry arises in molecules classified under VSEPR theory as AX₆E, where the central atom is surrounded by six bonding pairs and one lone pair, resulting in a pentagonal base with an apical ligand and the lone pair occupying an axial position in the electron geometry of pentagonal bipyramidal.^[37] This configuration leads to bond angles of approximately 72° in the equatorial plane and 90° between axial and equatorial positions, with the lone pair causing minimal distortion in stable cases.^[38] Representative examples include the xenon oxy pentafluoride anion, XeOF₅⁻, where xenon serves as the central atom bonded to one oxygen and five fluorines, exhibiting C₅ᵥ symmetry and polarity due to the asymmetric lone pair placement.^[37] Another instance is the iodate difluoride pentafluoride dianion, IOF₅²⁻, which adopts a similar arrangement with iodine at the core.^[38] In materials science, pentagonal pyramidal units model the assembly of virus capsids, particularly in icosahedral structures such as the T=3 Pariacoto virus, where each pentagonal face of an underlying dodecahedron serves as the base for a pyramid composed of capsid proteins, facilitating curvature and closure of the shell. This pyramidal motif, often rigid with a defined height and base radius, contributes to the T-number symmetry in viral architectures, enabling efficient self-assembly around nucleic acids.^[39] In nanotechnology, copper nanowires frequently exhibit pentagonal cross-sections due to five-fold twinning during growth, stabilizing atomic strands along the direction and enhancing mechanical strength compared to cubic structures.^[40] These nanowires, synthesized via aqueous reduction, display face-centered cubic packing within the pentagonal framework, with multi-shell variants incorporating folded {100} facets for improved conductivity.^[41] Quantum chemistry simulations employ pentagonal pyramidal coordinates to optimize energy minima in such systems, as seen in density functional theory calculations for novel ions like C₅H₆Si²⁺, where the geometry minimizes strain through precise apical-equatorial positioning.^[42] Historically, the 1960s marked key discoveries in coordination chemistry, with the synthesis of XeF₆ in 1962 sparking debates on its predicted pentagonal pyramidal structure under VSEPR, influencing models for higher coordination numbers despite its fluxional nature. This five-fold symmetry distinguishes pentagonal pyramids from octahedral-derived (square) pyramids by promoting unique σ- and π-bonding patterns, such as in transition metal complexes, that exploit equatorial delocalization unavailable in four-fold bases.^[37]

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Johnson Solid Near Misses
This near miss, discovered by Roger Kaufman in 2006, resembles the Johnson Solid 'bilunabirotunda ... pentagon replaced by a pentagonal pyramid. Only the ...<|control11|><|separator|>
[37]
https://doi.org/10.1016/j.jfluchem.2008.06.016
[38]
https://doi.org/10.1016/S0022-1139(99)00194-3
[39]
https://www.rpgroup.caltech.edu/virus_bootcamp/assets/pdfs/Bruinsma2006_Article_PhysicsOfRNAAndViralAssembly.pdf
[40]
Indication of Unusual Pentagonal Structures in Atomic-Size Cu ...
Sep 17, 2004 · Furthermore, Cu NW's with pentagonal cross section and axis along [110] direction have already been chemically synthesized [16] . Sen et al ...
[41]
Synthesis of Ultralong Copper Nanowires for High-Performance ...
Jul 19, 2012 · We report herein the novel synthesis of ultralong single-crystalline Cu nanowires with excellent dispersibility, providing an excellent candidate material
[42]
New Molecular Geometries With Pentagonal-Pyramidal Structure
Jul 6, 2025 · The new molecules with pentagonal-pyramidal geometry are C5H6Si2+, C5H5N2+, C5H5P2+, C5H6B+, and C5H6Al+, derived from benzene dication.Missing: examples | Show results with:examples