Hexagonal pyramid

A hexagonal pyramid is a three-dimensional polyhedron with a hexagonal base and six triangular lateral faces that meet at a common apex, forming a shape also known as a heptahedron.^[1] It features seven faces in total—one hexagonal base and six triangles—along with seven vertices (six on the base and one at the apex) and twelve edges (six forming the base and six connecting the base vertices to the apex).^[2] The lateral faces are typically isosceles triangles when the base is a regular hexagon, providing structural symmetry to the pyramid.^[3] Key properties of a hexagonal pyramid include its classification as a pyramid, where the apex is directly above the center of the base in the regular form, influencing its stability and geometric calculations.^[1] The volume of a hexagonal pyramid is calculated as V = a \times b \times h, where a is the apothem of the base, b is the side length of the base, and h is the height from the base to the apex; this formula derives from one-third of the base area multiplied by the height, with the base area being $3ab.^[3] The total surface area consists of the base area plus the lateral surface area, given by TSA = 3ab + 3bs, where s is the slant height of the triangular faces.^[1] These formulas assume a regular hexagonal base, though irregular variants exist with adjusted computations based on specific side lengths and angles.^[2]

Definition and classification

Definition

A pyramid is a polyhedron consisting of a polygonal base and triangular lateral faces that meet at a common vertex known as the apex.^[4] This structure distinguishes pyramids from prisms, which feature two parallel polygonal bases connected by rectangular or parallelogram lateral faces rather than converging at a single point.^[4] A hexagonal pyramid specifically features a hexagonal base—a polygon with six sides—and six triangular lateral faces that converge at the apex.^[5] The base may be regular, with equal sides and angles, or irregular, while the apex lies outside the plane of the base to form the pyramidal shape.^[6] Pyramids originated in ancient geometry as conceptual models for monumental structures, such as the stepped ziggurats of Mesopotamia, which served religious purposes and symbolized mountains reaching toward the heavens.^[7] These forms were later formalized within modern polyhedral geometry as part of the systematic classification of three-dimensional shapes.^[4]

As a Johnson solid

The regular hexagonal pyramid, when constructed with a regular hexagonal base and equilateral triangular lateral faces of equal edge length, does not qualify as a Johnson solid. Johnson solids are defined as strictly convex polyhedra composed of regular polygonal faces with all edges of equal length, excluding the five Platonic solids, the 13 Archimedean solids, infinite prisms, and infinite antiprisms; there are exactly 92 such solids.^[8] This exclusion arises from the systematic enumeration conducted by Norman W. Johnson in his 1966 Ph.D. thesis, where only the square pyramid (J1) and pentagonal pyramid (J2) appear among the simple pyramids satisfying the criteria.^[9] Higher n-gonal pyramids fail the conditions: for n=6, the configuration degenerates into a planar figure, while for n>6, no such convex 3D realization exists with regular faces and equal edges. (Note: The enumeration was published in Johnson's paper "Convex Polyhedra with Regular Faces" in the Canadian Journal of Mathematics, Vol. 18, No. 1 (1966), pp. 169–200.)^[10] The degeneracy for the hexagonal case stems from the geometry required for regularity. Consider a regular n-gon base with side length a; the distance from the center to a vertex is r = \frac{a}{2 \sin(\pi/n)}. For the lateral edges to also equal a, the pyramid height h satisfies \sqrt{h^2 + r^2} = a, so

h = a \sqrt{1 - \frac{1}{4 \sin^2 (\pi / n)}}.

Substituting n=6 yields \sin(\pi/6) = 1/2, so h = 0, collapsing the apex into the base plane and forming a non-convex, two-dimensional hexagon tiled by triangles rather than a strictly convex 3D polyhedron. This violates the convexity and volume requirements for Johnson solids, distinguishing the hexagonal pyramid from its square and pentagonal counterparts, which have positive heights (e.g., h \approx 0.707a for n=4). Thus, no hexagonal pyramid shares the face configuration of J1 or J2 in Johnson's catalog.

Structure

Faces, edges, and vertices

A hexagonal pyramid consists of 7 faces: one hexagonal base and 6 triangular lateral faces.^[5]^[4] The pyramid has 12 edges in total, comprising 6 edges along the hexagonal base and 6 additional edges connecting each base vertex to the apex.^[5]^[4] It possesses 7 vertices: 6 forming the base hexagon and 1 at the apex.^[5]^[4] These elements satisfy Euler's polyhedral formula, V - E + F = 7 - 12 + 7 = 2, verifying the structure as a convex polyhedron of genus 0.^[11] Regarding adjacency, each triangular lateral face shares one edge with the base and one edge with each of two neighboring triangular faces, while the apex vertex is incident to all 6 lateral edges.^[4]

Regular and irregular variants

The regular variant of a hexagonal pyramid features a base that is a regular hexagon and the apex positioned directly above the center of the base, resulting in six isosceles triangular lateral faces that provide symmetry and convexity.^[5] Irregular variants deviate from this ideal form in specific ways, such as an oblique hexagonal pyramid where the apex is displaced laterally from the perpendicular above the base center, resulting in non-equilateral triangular lateral faces that disrupt the symmetry.^[12]^[13] Another irregularity arises from a non-regular hexagonal base, where the sides and angles vary, leading to asymmetric lateral faces and reduced overall uniformity while still forming a pyramid.^[14]^[13] For a hexagonal pyramid to qualify as convex, its base must be a convex hexagon, and the apex must lie on the side of the base plane opposite to the interior, ensuring all internal angles are less than 180 degrees and the figure lies entirely on one side of any face plane.^[13] Non-convex variants, such as those with star-shaped (non-convex) bases, are typically excluded from standard pyramid classifications and instead considered stellated polyhedra.^[13] In applications, the regular hexagonal pyramid is favored in architectural modeling for its uniform symmetry, enabling precise structural designs like decorative spires or stable load-bearing forms.^[15] Irregular variants, particularly oblique or those with non-regular bases, better approximate natural formations such as pseudo-hexagonal pyramidal crystal structures observed in minerals like nepheline and zincite within the hexagonal crystal system.^[16]

Dimensions

Volume

The volume V of any 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.^[17] For a regular hexagonal pyramid, the base is a regular hexagon with side length s, and its area is A_b = \frac{3\sqrt{3}}{2} s^2.^[6] Substituting this into the general pyramid volume formula yields the specific volume for a regular hexagonal pyramid:

V = \frac{1}{3} \left( \frac{3\sqrt{3}}{2} s^2 \right) h = \frac{\sqrt{3}}{2} s^2 h,

assuming the apex is directly above the center of the base.^[5] This formula can be derived by considering the cross-sectional area parallel to the base at a distance x from the apex, which scales linearly with the square of the distance due to similar figures. The cross-sectional area at height x (measuring from the apex) is A(x) = A_b \left( \frac{x}{h} \right)^2, and integrating from x = 0 to x = h gives

V = \int_0^h A_b \left( \frac{x}{h} \right)^2 \, dx = A_b \frac{h}{3},

which simplifies to the general pyramid volume formula and thus to the hexagonal case upon substitution./06%3A_Applications_of_Integration/6.02%3A_Determining_Volumes_by_Slicing) The volume is expressed in cubic units consistent with the units of s and h. For example, with s = 1 and h = 1, V = \frac{\sqrt{3}}{2} \approx 0.866.^[5]

Surface area

The surface area of a regular hexagonal pyramid consists of the area of the hexagonal base and the lateral surface area formed by the six isosceles triangular faces meeting at the apex. The base area is calculated as the area of a regular hexagon with side length s, which is \frac{3\sqrt{3}}{2} s^2.^[5] The lateral surface area is determined using the slant height l, defined as the perpendicular distance from the midpoint of a base edge to the apex along the face of the pyramid. This slant height is derived from the Pythagorean theorem applied to the right triangle formed by the pyramid's height h, the apothem of the base (the distance from the center to the midpoint of a side, \frac{s \sqrt{3}}{2}), and the slant height itself: l = \sqrt{h^2 + \left( \frac{s \sqrt{3}}{2} \right)^2 } = \sqrt{h^2 + \frac{3 s^2}{4} }. ^[5] The lateral surface area is then the product of half the base perimeter (which is $3s) and the slant height: $3 s l, assuming the triangular faces are isosceles, as is standard for a regular pyramid.^[4] This follows the general formula for the lateral surface area of a pyramid, \frac{1}{2} \times perimeter \times l, where the perimeter of the hexagonal base is $6s.^[18] The total surface area (TSA) is the sum of the base area and the lateral surface area: \text{TSA} = \frac{3\sqrt{3}}{2} s^2 + 3 s l. Substituting the expression for l yields the closed-form \text{TSA} = \frac{3}{2} s \left( s \sqrt{3} + \sqrt{3 s^2 + 4 h^2} \right). ^[5] Surface areas are expressed in square units corresponding to the units of s and h. For example, with s = 1 and h = 1, l \approx 1.3229, the base area is approximately 2.598, the lateral surface area is approximately 3.969, and the total surface area is approximately 6.567.^[18]

Coordinates

Cartesian coordinates

A regular hexagonal pyramid is commonly positioned in Cartesian coordinates with its hexagonal base centered at the origin in the xy-plane and the apex located directly above the center along the positive z-axis. This placement ensures the pyramid is right and regular, with the base vertices forming a regular hexagon and the lateral edges connecting to the apex. For a base side length of s = 1, the six base vertices are given by:

(1, 0, 0),\ \left(\frac{1}{2}, \frac{\sqrt{3}}{2}, 0\right),\ \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}, 0\right),\ (-1, 0, 0),\ \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}, 0\right),\ \left(\frac{1}{2}, -\frac{\sqrt{3}}{2}, 0\right).

[(https://web.mae.ufl.edu/uhk/NSIDED-POLYGONS.pdf)] The apex is at (0, 0, h), where h > 0 is the height of the pyramid. To obtain coordinates for a general base side length s, the x- and y-coordinates of the base vertices are multiplied by s, while the z-coordinates remain unchanged; this scaling preserves the distance of s between adjacent base vertices.[(https://web.mae.ufl.edu/uhk/NSIDED-POLYGONS.pdf)] This coordinate system defines exactly 7 vertices for the pyramid: the 6 base points and the apex. Using these points, quantities such as the slant height (via Euclidean distances between vertices) or volume (via integration or determinant formulas over the tetrahedra formed by the apex and base triangles) can be computed. Such explicit coordinates facilitate applications in computer graphics for rendering the pyramid, 3D printing to generate STL files from vertex data, and finite element analysis for meshing the structure in simulations.