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Cuboid

A cuboid is a three-dimensional polyhedron with six rectangular faces, twelve edges, and eight vertices, where all angles are right angles and opposite faces are congruent and parallel.^[1] Also known as a rectangular prism or rectangular parallelepiped, it is a special case of a more general hexahedron in which the faces are rectangles rather than arbitrary quadrilaterals.^[2] The shape is defined by three pairwise perpendicular dimensions: length (l), width (w), and height (h), which can vary independently to form rectangular boxes of different proportions.^[3] Cuboids are fundamental in geometry and appear ubiquitously in everyday objects and architecture, such as books, bricks, shipping containers, and building blocks, due to their structural stability and efficient space utilization.^[4] Key properties include the fact that all edges meet at 90-degree angles, ensuring orthogonality, and the faces lie in three pairs of parallel planes.^[5] The volume of a cuboid is calculated as V = l × w × h, representing the space it encloses, while its surface area is A = 2(lw + lh + wh), useful for determining material needs in manufacturing or packaging.^[6] In coordinate geometry, a cuboid can be aligned with the axes for simplicity, with vertices at points like (0,0,0) and (l,w,h), facilitating computations in fields like computer graphics and engineering.^[1] Unlike a cube, which is a regular cuboid where all dimensions are equal, general cuboids allow for rectangular asymmetry, broadening their applications in design and physics simulations.^[2]

Definition and Basic Properties

Definition

A cuboid is a convex hexahedron, defined as a polyhedron with six faces, bounded by six rectangular faces, twelve edges, and eight vertices, where all faces are rectangles and all edges meet at right angles.^[1]^[3] This structure ensures that opposite faces are equal in area and parallel, forming a right prism with rectangular bases.^[7] The term "cuboid" is also used interchangeably with "rectangular parallelepiped" or "rectangular box," emphasizing its orthogonal geometry.^[1] Unlike a general parallelepiped, which features six parallelogram faces that may be oblique, a cuboid specifically requires right angles at all vertices, distinguishing it as the orthogonal special case. The concept assumes familiarity with basic polygons, such as rectangles, and polyhedra as three-dimensional solids with flat polygonal faces, though the key term hexahedron is defined here as any polyhedron with exactly six faces.^[8] Etymologically, "cuboid" derives from the Latin "cubus" (cube) combined with the suffix "-oid," indicating resemblance to a cube; it first appeared in English mathematical texts in the early 19th century, with the earliest recorded use in 1829.^[9]

Faces, Edges, and Vertices

A cuboid possesses six faces, each a rectangle, forming three pairs of congruent opposite faces. These pairs are mutually perpendicular, with the faces in each pair parallel to one another and the edges where adjacent faces meet forming right angles.^[10]^[11] The cuboid features twelve edges, organized into three groups of four parallel edges each, where the edges within a group share identical lengths that define the three distinct dimensions of the shape—typically length, width, and height.^[10] At its eight vertices, three edges converge at right angles, establishing the orthogonal connectivity that characterizes the cuboid's framework.^[11]^[12] This arrangement yields a skeletal structure where each vertex links to exactly three others via these edges, underscoring the shape's rigid, grid-like topology without any oblique intersections. As a convex polyhedron, the cuboid adheres to Euler's formula, V - E + F = 8 - 12 + 6 = 2, where V denotes vertices, E edges, and F faces, thereby confirming its topological integrity.^[13] This orthogonality pervades the entire form, ensuring all dihedral angles are 90 degrees and eliminating any potential for skewed alignments.^[10]

Dimensions and Measurements

Volume

The volume of a cuboid, which quantifies the three-dimensional space it encloses, is given by the formula V = l \times w \times h, where l, w, and h denote the lengths of the three edges meeting at any vertex, representing the dimensions along each axis.^[1] This formula derives from conceptualizing the cuboid as a prism with a rectangular base of area l \times w, extruded by the height h; thus, the total volume is the base area multiplied by the height, yielding V = (l \times w) \times h. Equivalently, by Cavalieri's principle, the volume equals the integral of cross-sectional areas parallel to the base over the height interval; for a cuboid, each such cross-section is a rectangle of constant area l \times w, so the integral simplifies to (l \times w) \times h.^[14] The unit of volume is cubic, matching the units of the input dimensions—for instance, if l, w, and h are measured in meters, then V is in cubic meters (m³).^[15] The volume scales cubically with linear dimensions: it is proportional to the product lwh, so tripling any single dimension triples the volume, while tripling all three multiplies it by 27.^[15] As an illustrative example, consider a cuboid with dimensions l = 3 units, w = 2 units, and h = 1 unit; its volume is V = 3 \times 2 \times 1 = 6 cubic units.^[1] In the special case where l = w = h = s, the cuboid reduces to a cube with volume V = s^3.

Surface Area

The total surface area of a cuboid, which encompasses the external area of all six rectangular faces, is calculated by summing the areas of three pairs of identical opposite faces. Each pair consists of two faces with dimensions l × w, l × h, and w × h, where l is the length, w is the width, and h is the height. Thus, the formula is SA = 2(lw + lh + wh) square units.^[16]^[17] This formula derives from the geometric structure of the cuboid, where unfolding the solid into a two-dimensional net reveals six rectangles arranged such that opposite faces match in area. The contribution from each pair is twice the product of the two relevant dimensions, ensuring the total accounts for the complete boundary without overlap.^[16]^[18] The lateral surface area, excluding the top and bottom bases (assumed to be l × w), focuses on the four side faces and is given by LSA = 2h(l + w) square units. This measure is particularly relevant for right prisms, where the height is perpendicular to the base.^[16]^[17] Surface area is expressed in square units, such as square meters (m²) or square centimeters (cm²), reflecting its two-dimensional nature. For a fixed volume, the surface area of a cuboid is minimized when all dimensions are equal, reducing it to a cube.^[17]^[19] For example, consider a cuboid with l = 3 units, w = 2 units, and h = 1 unit. The total surface area is SA = 2(3 \times 2 + 3 \times 1 + 2 \times 1) = 2(6 + 3 + 2) = 22 square units.^[16]

Space Diagonal

The space diagonal of a cuboid is a line segment that connects two non-adjacent vertices passing through the interior of the solid. For a cuboid with dimensions length l, width w, and height h, the length d of each space diagonal is calculated using the formula

d = \sqrt{l^2 + w^2 + h^2}.

This expression applies the three-dimensional Pythagorean theorem to the right-angled structure of the cuboid.^[7] To derive the formula, apply the Pythagorean theorem stepwise. First, the diagonal across the base face (spanning l and w) has length \sqrt{l^2 + w^2}. The space diagonal then forms the hypotenuse of a right triangle with this face diagonal as one leg and the height h as the other leg, yielding

d = \sqrt{ \left( \sqrt{l^2 + w^2} \right)^2 + h^2 } = \sqrt{l^2 + w^2 + h^2}.

This derivation holds regardless of which pair of opposite faces is chosen as the base, confirming the formula's consistency. In a cuboid, there are exactly four space diagonals, each connecting a pair of opposite vertices, and all have equal length due to the parallelism and perpendicularity of the edges.^[7] The space diagonal is expressed in the same linear units as the cuboid's dimensions, such as meters or centimeters. For instance, with l = 3, w = 2, and h = 1, the space diagonal is d = \sqrt{9 + 4 + 1} = \sqrt{14} \approx 3.74.^[7] The space diagonal represents the longest straight-line distance between any two vertices in the cuboid.^[7]

Mathematical Representation

In Cartesian Coordinates

A cuboid is commonly represented in three-dimensional Cartesian coordinates by aligning it with the x-, y-, and z-axes, which simplifies its mathematical description and computations. In this standard axis-aligned placement, the cuboid is defined by its dimensions: length l along the x-axis, width w along the y-axis, and height h along the z-axis. One vertex is positioned at the origin (0, 0, 0), and the opposite vertex is at (l, w, h). The coordinates of all eight vertices are then the combinations of these bounds: (0,0,0), (l,0,0), (0,w,0), (l,w,0), (0,0,h), (l,0,h), (0,w,h), and (l,w,h).^[20] For a more general position without rotation, the cuboid can be translated by a vector (x_0, y_0, z_0), shifting all vertices accordingly: (x_0, y_0, z_0), (x_0 + l, y_0, z_0), (x_0, y_0 + w, z_0), (x_0 + l, y_0 + w, z_0), (x_0, y_0, z_0 + h), (x_0 + l, y_0, z_0 + h), (x_0, y_0 + w, z_0 + h), and (x_0 + l, y_0 + w, z_0 + h). While rotations are possible to orient the cuboid arbitrarily in space, the axis-aligned configuration is emphasized here due to its orthogonality and prevalence in applications, where edge directions remain parallel to the coordinate axes.^[21] The six faces of the axis-aligned cuboid lie on planes parallel to the coordinate planes. For the standard placement, these are given by the equations x = 0, x = l, y = 0, y = w, z = 0, and z = h. After translation by (x_0, y_0, z_0), the plane equations become x = x_0, x = x_0 + l, y = y_0, y = y_0 + w, z = z_0, and z = z_0 + h.^[22] In computer graphics and computational geometry, the cuboid often serves as an axis-aligned bounding box (AABB), which encloses a set of points, models, or objects by taking the minimum and maximum extents in each coordinate direction. The vertices of such an AABB are generated as all possible combinations of the minimum (x_{\min}, y_{\min}, z_{\min}) and maximum (x_{\max}, y_{\max}, z_{\max}) coordinates, forming a rectangular prism that tightly bounds the enclosed geometry for efficient collision detection and visibility culling.

Parametric Equations

Parametric representations of a cuboid allow for modeling its surfaces and interior using continuous parameters, facilitating applications in integration, computer graphics, and geometric analysis. For the surfaces, each of the six rectangular faces can be parameterized individually using two variables. Consider a cuboid aligned with the coordinate axes, with one vertex at the origin, length l along the x-axis, width w along the y-axis, and height h along the z-axis. The bottom face in the xy-plane is parameterized as

\vec{r}(u, v) = (u, v, 0), \quad 0 \leq u \leq l, \ 0 \leq v \leq w.

This bilinear mapping stretches a unit square in the uv-plane to the face dimensions. The top face is similarly given by \vec{r}(u, v) = (u, v, h) with the same parameter ranges. The four side faces follow analogous forms: for the front face, \vec{r}(u, v) = (u, 0, v) for $0 \leq u \leq l, $0 \leq v \leq h; the back face by \vec{r}(u, v) = (u, w, v); the left face by \vec{r}(u, v) = (0, v, u); and the right face by \vec{r}(u, v) = (l, v, u), adjusting ranges accordingly. To parameterize the entire hexahedral surface, these six charts are pieced together, with parameters scaled to the respective dimensions, ensuring continuity at edges where adjacent faces meet. This approach is common in surface integral computations, where the parameterization enables evaluation over each face separately. For the interior volume, a three-parameter representation maps the unit cube to the cuboid:

\vec{r}(s, t, u) = (s l, t w, u h), \quad 0 \leq s, t, u \leq 1.

This affine transformation preserves the rectangular structure and is particularly useful in triple integrals for volume calculations, as the Jacobian determinant equals l w h, confirming the volume as l w h times the unit cube volume. In vector form, assuming alignment with the orthogonal basis, points within the cuboid are expressed as \vec{r} = s \vec{a} + t \vec{b} + p \vec{c}, where \vec{a} = l \hat{i}, \vec{b} = w \hat{j}, \vec{c} = h \hat{k} are the edge vectors from the origin, and $0 \leq s, t, p \leq 1. This linear combination generalizes the parameterization for affine transformations of the unit cube into the cuboid.^[23]

Parallelepiped

A parallelepiped is a three-dimensional polyhedron with six faces, each a parallelogram, formed by three pairs of identical, parallel faces. Unlike the cuboid, where all faces are rectangles and edges are perpendicular, the edges of a parallelepiped occur in three sets of four parallel lines but are not necessarily at right angles to each other. The cuboid represents a special orthogonal case of this more general figure.^[24]^[25] Parallelepipeds are classified by the shape of their faces, including the rectangular type (equivalent to the cuboid, with right-angled parallelograms), the rhombohedral type (all faces as rhombi with equal edge lengths but non-right angles), and the general oblique type (arbitrary parallelograms with varying angles and side lengths). In all cases, opposite faces are congruent and parallel, ensuring the structure's uniformity along each direction. Key properties include non-rectangular faces in the general form, where each parallelogram face has two diagonals of potentially unequal lengths, and four space diagonals connecting opposite vertices that intersect at the figure's centroid but differ in length. The diagonals of a parallelepiped intersect at one point and are bisected by this point.^[26] The volume of a parallelepiped differs from the cuboid's length-width-height product, instead relying on the orientation of its edges; it is calculated as the absolute value of the scalar triple product of three vectors \mathbf{a}, \mathbf{b}, and \mathbf{c} representing edges from one vertex:

V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|.

This formula captures the base area times height, accounting for obliqueness via the cross product's magnitude and the dot product's projection. A cuboid can be converted to a general parallelepiped by applying a shearing transformation, which tilts the faces parallel to a chosen plane while maintaining parallelism and volume.^[27]^[28]^[29]

Cube as Special Case

A cube is a special case of a cuboid in which all three dimensions are equal, denoted as length = width = height = a, where a is the side length.^[30] In this configuration, the cuboid becomes a regular hexahedron with all six faces as congruent squares, all twelve edges of equal length a, and all four space diagonals of equal length. This equality imparts the cube with the highest degree of symmetry among cuboids, governed by the full octahedral group O_h of order 48, which includes both proper rotations and improper isometries such as reflections and inversions.^[31] The formulas for the cube simplify directly from the general cuboid expressions by setting all dimensions to a. The volume is V = a^3, representing the space enclosed by the six square faces.^[30] The total surface area reduces to SA = 6a^2, as each of the six faces has area a^2.^[32] The length of each space diagonal, connecting opposite vertices through the interior, is d = a\sqrt{3}.^[33] The isometries of the cube consist of 24 proper rotations—such as those around axes through vertices, face centers, or edge midpoints—and 24 improper isometries, including reflections across planes through edges or face diagonals and rotary inversions, all of which map the cube onto itself while preserving distances.^[34] These transformations form the complete symmetry group, highlighting the cube's uniformity compared to general cuboids with lower symmetry.^[35] Historically, the cube predates the formal terminology of the cuboid, with references appearing in ancient geometry; in Plato's Timaeus, the cube is associated with the element earth due to its stability and right-angled form, one of five regular polyhedra linked to the classical elements.^[36]

Applications

In Packaging and Storage

Cuboids form the basis of standard rectangular packaging, such as cardboard boxes, which are ubiquitous for containing and transporting consumer goods. These boxes are designed with dimensions optimized to maximize internal volume while minimizing material usage, often by reducing surface area for a given volume to lower production costs and waste.^[37] For example, manufacturers calculate box sizes to achieve the best volume-to-surface-area ratio, ensuring efficient containment without excess cardboard.^[38] The geometric properties of cuboids enable perfect tessellation in three-dimensional space, allowing them to stack seamlessly without gaps or overlaps, which is critical for warehouse storage efficiency. This capability supports high-density palletizing and racking systems, optimizing storage cube utilization (the percentage of available storage volume occupied by inventory) in logistics facilities to between 22% and 27% of total capacity as an ideal benchmark, balancing accessibility for picking and restocking.^[39] Volume and surface area serve as key metrics in assessing this stacking performance, directly influencing storage throughput and operational costs.^[40] Shipping containers exemplify large-scale cuboid applications, adhering to ISO 668 standards for interoperability in global trade. The 20-foot container, a common variant, has external dimensions of approximately 6.1 m in length, 2.4 m in width, and 2.6 m in height, with an internal volume of about 33 m³, enabling standardized loading on ships, trucks, and trains.^[41] From an economic perspective, manufacturing costs for cuboid packaging are determined by factors tied to volume, which dictates capacity and payload, and surface area, which influences material and printing expenses. These calculations typically include raw material procurement, labor for assembly, and tooling setup, with surface area directly scaling the amount of corrugated board or other substrates required per unit.^[42] Optimization here reduces overall expenses, as smaller surface areas for equivalent volumes can cut material needs by up to 30% in efficient designs.^[37] Environmentally, cuboid packaging benefits from recyclable materials like cardboard, which is derived from renewable fibers and achieves recycling rates exceeding 80% in many regions, thereby minimizing landfill waste and carbon emissions compared to plastic alternatives.^[43] This recyclability supports sustainable logistics by enabling closed-loop systems where used boxes are repurposed, reducing the environmental footprint of packaging production.^[44]

In Architecture and Engineering

In architecture, cuboids form the foundational building blocks of many structures, particularly through materials like bricks and concrete forms, which are standardized into rectangular prism shapes to ensure efficient stacking and alignment. Bricks, typically measuring around 215 mm × 102.5 mm × 65 mm in standard imperial sizes, leverage their cuboid geometry for optimal load distribution in masonry walls, where the right-angled faces allow for perpendicular bonding that enhances compressive strength without requiring complex interlocking.^[45] Similarly, concrete blocks, often cast in cuboid molds of dimensions such as 390 mm × 190 mm × 190 mm, facilitate rapid formwork assembly and provide uniform load-bearing capacity in foundational and partition elements, contributing to the structural integrity of low- to mid-rise buildings.^[46] This geometric simplicity minimizes material waste and supports vertical load transfer, making cuboids ideal for scalable construction in regions with seismic considerations, where right angles help predict and mitigate shear stresses. Interior spaces in modern architecture frequently adopt cuboid configurations for rooms and enclosures, simplifying the design and calculation of essential systems like heating, ventilation, and air conditioning (HVAC) as well as lighting. The orthogonal dimensions of a cuboid room—length, width, and height—enable straightforward volumetric computations for airflow rates and thermal loads in HVAC systems, often using formulas based on room area and ceiling height to determine required cubic feet per minute (CFM) of air exchange.^[47] For lighting, architects calculate illuminance levels in lux by considering the planar surfaces of cuboid interiors, which allow for uniform distribution assumptions in simulations, ensuring compliance with standards like those from the Illuminating Engineering Society that recommend 300–500 lux for office spaces.^[48] This approach not only streamlines energy modeling but also optimizes fixture placement along parallel walls and ceilings, reducing shadows and enhancing occupant comfort in commercial and residential designs. Exemplifying the cuboid's prominence in modernist architecture, the Seagram Building in New York City, completed in 1958 by Ludwig Mies van der Rohe and Philip Johnson, embodies the rectangular slab form as a hallmark of International Style purity, rising 515 feet in a bronzed I-beam grid that emphasizes structural honesty and verticality.^[49] In engineering applications, finite element analysis (FEA) is routinely applied to cuboid components to evaluate stress concentrations, particularly along edges where tensile forces can lead to failure under dynamic loads; for instance, simulations of cuboid lattice structures reveal vibration modes influenced by edge reinforcements, informing designs in aerospace and civil engineering.^[50] Standardization further amplifies the cuboid's utility in modular construction, where prefabricated units adhere to interface guidelines like those in emerging building codes, enabling scalable assembly of multi-story frames with precise tolerances for alignment and expansion.^[51] These practices underscore the cuboid's role in balancing aesthetic minimalism with engineering precision, as seen in parametric optimizations of cuboid building volumes for energy efficiency.^[52]

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AD Classics: Seagram Building / Mies van der Rohe - ArchDaily
May 10, 2010 · Located in the heart of New York City, the Seagram Building designed by Mies van der Rohe epitomizes elegance and the principles of modernism.Missing: rectangular form
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FE vibration analyses of novel conforming meta-structures ... - Nature
Dec 8, 2020 · In Part 1, a simple rectangular cuboid (50 × 50 × 100 mm) solid is used as a basis for comparing vibration responses of varying density lattices ...
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Modular Construction Interface Standard Aims to Establish Clear ...
Apr 10, 2025 · A new building code standard hopes to create guidelines for the connection and configuration of modular building components.<|control11|><|separator|>
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https://www.mdpi.com/2075-5309/12/1/69