Rectangular cuboid
A rectangular cuboid, also known as a rectangular prism or simply a cuboid, is a three-dimensional geometric solid with six rectangular faces, twelve edges, and eight vertices, where all interior angles are right angles (90 degrees).[1][2][3] Opposite faces of a rectangular cuboid are congruent and parallel, and its edges meet at right angles, distinguishing it from more general polyhedra like oblique prisms.[4][5] This shape is fundamental in geometry and appears ubiquitously in everyday objects and architecture, such as bricks, books, shipping containers, and building foundations, due to its structural stability and efficient space utilization.[1][6] The volume of a rectangular cuboid is calculated as the product of its three dimensions—length, width, and height—while its surface area is the sum of the areas of its six faces, typically expressed as $2(lw + lh + wh), where l, w, and h denote the respective dimensions.[7][8] In coordinate geometry, a rectangular cuboid can be defined by two opposite vertices in three-dimensional space, with edges aligned parallel to the axes for simplicity in computations.[9] Rectangular cuboids play a key role in fields like engineering, physics, and computer graphics, where they model bounded regions for simulations, such as collision detection in video games or material stress analysis in design.[7] Their orthogonal properties facilitate straightforward mathematical operations, including scaling, translation, and integration over their volume for applications in calculus and optimization problems.[4]Definition and Terminology
Formal Definition
A rectangular cuboid is a convex polyhedron bounded by six pairwise parallel rectangles, forming a three-dimensional solid where all face angles are 90 degrees and all edges meet perpendicularly at the vertices.[10] This structure ensures that the figure is a special type of hexahedron, with opposite faces being congruent rectangles and all dihedral angles measuring exactly 90 degrees.[11] Unlike general polyhedra or irregular hexahedra, which may have non-rectangular faces and oblique angles, the rectangular cuboid requires all faces to be rectangles with right angles, distinguishing it as a right rectangular prism or rectangular parallelepiped.[10] This prerequisite emphasizes the orthogonality of its bounding planes, where the three pairs of identical rectangular faces are aligned along mutually perpendicular axes, providing a rigid geometric framework.[11] The concept of the rectangular cuboid originates from Euclidean geometry, as explored in Book XI of Euclid's Elements (c. 300 BCE), where parallelepipedal solids are analyzed in the context of solid geometry, with the rectangular cuboid as a special case featuring right angles.[12] It was further formalized in the 17th century through René Descartes' development of analytic geometry in La Géométrie (1637), which introduced coordinate systems to precisely describe such orthogonal solids algebraically. The cube represents a special case of the rectangular cuboid in which all edges are equal in length.[10]Common Names and Variations
The rectangular cuboid is referred to by several synonymous terms in mathematical and geometric contexts, including rectangular prism, rectangular parallelepiped, and right prism.[10][13] In higher-dimensional geometry, the term orthotope serves as a generalization, describing a parallelotope with mutually perpendicular edges that extends the concept of the rectangular cuboid beyond three dimensions.[14] These synonyms emphasize the shape's defining characteristics of six rectangular faces meeting at right angles. While the term "cuboid" broadly denotes a hexahedron bounded by six quadrilateral faces in some geometric definitions, the rectangular cuboid specifically requires all faces to be rectangles, ensuring all dihedral angles are 90 degrees and distinguishing it from oblique variants like the general parallelepiped, which may have slanted edges.[10] This precision avoids confusion with irregular hexahedra, where faces are not necessarily parallelograms; here, the rectangular specification confines the shape to orthogonal alignments, aligning with its role as a right cuboid.[15] The etymology of "cuboid" traces to the Greek "kybos," meaning a six-sided die or cube, combined with the suffix "-oid" from "-oeidēs," indicating resemblance or likeness in form, thus underscoring its cube-like structure but allowing for unequal dimensions.[16] This nomenclature highlights the shape's foundational resemblance to the cube while accommodating variations in edge lengths. In practical applications, the rectangular cuboid is commonly called a "box" in everyday language, evoking familiar objects like storage containers or rooms.[13] In architecture, elongated forms are often termed "bricks," which are standardized cuboids used for constructing walls and structures due to their efficient stacking properties.[17]Geometric Properties
Faces, Edges, and Vertices
A rectangular cuboid consists of six rectangular faces, which form the bounding surfaces of the solid. These faces are grouped into three pairs of congruent and parallel opposites, typically referred to as the front and back, left and right, and top and bottom, with each pair sharing identical dimensions.[10] The cuboid features twelve straight edges that define the boundaries between adjacent faces. These edges are organized into four equal lengths along each of the three dimensions—length, width, and height—with all edges meeting at right angles to ensure the rectangular configuration.[10] At the intersections of these edges lie eight vertices, each connecting exactly three edges to form the corners of the cuboid. The connectivity of these vertices and edges constitutes a graph that is isomorphic to the cubical graph, the skeleton of a cube, although the edge lengths in a general rectangular cuboid may vary across the three dimensions.[18][10] In terms of adjacency, each rectangular face shares one edge with each of four adjacent faces, creating a closed polyhedral structure with no curved surfaces or non-planar faces, all aligned along three mutually perpendicular directions.[10][19]Symmetry and Angles
The rectangular cuboid exhibits perfect orthogonality in its angular structure, with all dihedral angles—the angles between adjacent faces—measuring exactly 90 degrees, ensuring that the six rectangular faces meet perpendicularly at each edge.[10] Within each face, the four interior angles are also 90 degrees, consistent with the rectangular geometry of the faces.[20] This right-angled configuration distinguishes the rectangular cuboid from more general parallelepipeds, where dihedral angles may deviate from orthogonality.[21] The inherent symmetry of the rectangular cuboid is captured by its point group, denoted as D_{2h} in Schoenflies notation, which encompasses 8 symmetry operations: the identity element, three twofold rotations (180 degrees) about the three principal axes aligned with the edges, the inversion center, and three mirror reflections through the planes bisecting the coordinate axes. This group reflects the cuboid's orthorhombic symmetry, featuring three mutually perpendicular twofold rotation axes without higher-order rotations.[22] In contrast, a cube possesses the fuller octahedral group O_h with 48 symmetry elements, including rotations of various orders and additional reflections; however, when the cuboid's three edge lengths are unequal, the loss of equal dimensions reduces the symmetry order from 48 to 8, eliminating operations that would interchange unequal axes. A defining feature of this symmetry arises from the alignment of the cuboid's 12 edges parallel to three mutually perpendicular axes, which facilitates the decomposition of any vector within the cuboid into orthogonal components along these directions and underpins the D_{2h} operations.[5] These axes connect the cuboid's 8 vertices and midpoints of opposite faces or edges, serving as the loci for the group's rotational and reflective symmetries.Measurements and Formulas
Volume Calculation
The volume V of a rectangular cuboid, defined by three mutually perpendicular edge lengths l, w, and h, is calculated using the formula V = l \times w \times h. This expression quantifies the space enclosed within the six rectangular faces, representing a fundamental measure in three-dimensional geometry. The result is expressed in cubic units consistent with the input dimensions, such as cubic meters (m³) when l, w, and h are measured in meters. The formula derives from the geometric principle that the enclosed space equals the area of a rectangular base (l \times w) extruded perpendicularly by the height h, akin to filling the shape with unit cubes along each dimension.[23] This approach aligns with Cavalieri's principle, which states that two solids sharing the same height and identical cross-sectional areas at every level have equal volumes; for a rectangular cuboid, the constant rectangular cross-sections parallel to the base confirm the product formula holds equivalently for prisms with the same base and height.[24] In this context, the rectangular cuboid is a right prism with rectangular bases, ensuring uniform cross-sections.[25] A rigorous proof via coordinate geometry positions one vertex of the cuboid at the origin (0,0,0), with edges aligned along the coordinate axes to the points (l,0,0), (0,w,0), and (0,0,h). The volume is then the triple integral of the constant function 1 over the bounded region R: \begin{align*} V &= \iiint_R \, dx \, dy \, dz \\ &= \int_0^l \int_0^w \int_0^h 1 \, dz \, dy \, dx \\ &= \int_0^l \int_0^w h \, dy \, dx \\ &= \int_0^l h w \, dx \\ &= l w h. \end{align*} This scaling reflects the cuboid's dependence on the three perpendicular dimensions described in its geometric properties. Additionally, if all edge lengths are scaled uniformly by a positive factor k, the volume transforms as V' = k^3 V, preserving the proportional enclosure of space under similarity transformations.[26] In the special case of a cube, where l = w = h = a, the formula simplifies to V = a^3, emphasizing the isotropic nature of this equilateral cuboid.[23]Surface Area and Diagonals
The surface area of a rectangular cuboid, with dimensions length l, width w, and height h, is calculated as the sum of the areas of its six rectangular faces, consisting of three pairs of identical opposite faces: SA = 2(lw + lh + wh). This formula arises from adding twice the area of each unique face pair—the two l \times w faces, the two l \times h faces, and the two w \times h faces.[10] A rectangular cuboid has 12 face diagonals, with two on each of its six faces. The length of a face diagonal on a face with sides a and b is d_{\text{face}} = \sqrt{a^2 + b^2}, applying the Pythagorean theorem to the right triangle formed by the sides and diagonal within that face. Thus, there are three distinct face diagonal lengths: \sqrt{l^2 + w^2} for the top and bottom faces, \sqrt{l^2 + h^2} for the front and back faces, and \sqrt{w^2 + h^2} for the side faces.[10] The space diagonals of a rectangular cuboid connect opposite vertices through the interior, spanning all three dimensions. There are four such diagonals, each with length d_{\text{space}} = \sqrt{l^2 + w^2 + h^2}, obtained by extending the Pythagorean theorem to three dimensions: first forming a face diagonal and then combining it with the third edge perpendicular to that face. In vector terms, if the edge vectors are \vec{l}, \vec{w}, and \vec{h} (mutually orthogonal), a space diagonal vector is \vec{l} + \vec{w} + \vec{h}, with magnitude \|\vec{l} + \vec{w} + \vec{h}\| = \sqrt{l^2 + w^2 + h^2}; face diagonals similarly use the sum of two edge vectors.[10][27] The total length of the 12 edges is $4(l + w + h), comprising four edges of each dimension. This metric provides a structural measure of the cuboid's framework.Coordinate Geometry
Cartesian Representation
In the Cartesian coordinate system, a rectangular cuboid is typically aligned with the coordinate axes for simplicity in representation and computation. One standard placement positions one vertex at the origin (0, 0, 0) and the opposite vertex at (l, w, h), where l, w, and h represent the edge lengths along the x-, y-, and z-axes, respectively. This alignment ensures that all edges are parallel to the axes, facilitating straightforward parameterization and calculations in Euclidean 3D space.[10][28] The eight vertices of the cuboid are formed by all possible combinations from the sets \{0, l\} \times \{0, w\} \times \{0, h\}. These vertices are:- (0, 0, 0)
- (l, 0, 0)
- (0, w, 0)
- (l, w, 0)
- (0, 0, h)
- (l, 0, h)
- (0, w, h)
- (l, w, h)