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Cuboid: definition, geometry, formulas, and applications

A cuboid is a six-faced polyhedron whose faces are quadrilaterals; rectangular cuboids (right rectangular prisms) and the cube are important special cases with widely used volume and surface formulas.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: April 19, 2026

simple.wikipedia.org · CC BY-SA 3.0

Definition and basic idea

A cuboid is a three-dimensional solid bounded by six quadrilateral faces. In common usage a cuboid refers to a right rectangular prism whose faces are rectangles; a special case in which all six faces are squares is the cube. More formally, a cuboid is a convex polyhedron made up of three pairs of congruent, parallel faces. The term emphasizes a cube-like, box-shaped geometry and is often used to describe everyday objects such as boxes, bricks and rooms.

Geometry and primary characteristics

Typical characteristics of a cuboid include:

Faces: 6
Edges: 12
Vertices (corners): 8
Face shapes: usually rectangles for a rectangular cuboid; more generally any quadrilaterals could form a convex six-faced solid, but such shapes are typically classified under different names.

When all adjacent faces meet at right angles it is called a rectangular cuboid or right rectangular prism. In geometry courses the cuboid is often introduced as the three-dimensional analogue of a rectangle: compare the 2D rectangle to the 3D cuboid.

Formulas and measurements

Let length = l, breadth (width) = b and height = h. Standard measures for any rectangular cuboid are:

Volume: V = l × b × h. This gives how much space the solid encloses (units³).
Total surface area (TSA): TSA = 2(l×b + b×h + h×l). This sums the areas of all six rectangular faces (units²).
Face diagonals: diagonal of face with sides l and b is √(l² + b²); similarly for other faces.
Space (body) diagonal: the longest straight line between two opposite vertices is √(l² + b² + h²).

These formulas follow from basic Euclidean geometry: areas of rectangles and the Pythagorean theorem for diagonals. When giving numerical answers, include appropriate units and, if necessary, round according to context.

Relations, distinctions and representations

A rectangular cuboid is a specific type of prism (a right prism with a rectangular base). It differs from a parallelepiped in that the latter may have parallelogram faces that are not right-angled. Another close term is "box" or "brick" in practical contexts. In coordinate geometry a cuboid aligned with the axes can be described as the set of points (x,y,z) satisfying x₁ ≤ x ≤ x₂, y₁ ≤ y ≤ y₂, z₁ ≤ z ≤ z₂. A cuboid can also be unfolded into a net — for the cube there are 11 distinct nets — which helps visualize all faces as a single planar figure.

History, naming and notable facts

The concept of block-shaped solids appears throughout classical geometry and practical construction. The English word "cuboid" derives from cube + -oid ("cube-like"). In mathematical literature one often encounters the terms rectangular solid, right rectangular prism and box interchangeably when the faces are rectangles. For introductions to polyhedra and three-dimensional solids see general references on polyhedra and convex solids (polyhedron overview).

Applications and examples

Cuboids model many everyday objects and engineered components: shipping containers, rooms, bricks, cereal boxes and many mechanical parts. They are used in packing problems, architectural planning, manufacturing and computer graphics (axis-aligned bounding boxes and voxels). Educationally, cuboids provide straightforward examples for teaching volume, surface area, and spatial reasoning. For further reading and interactive explanations, consult introductory geometry resources and educational pages on prisms and solids (quadrilateral faces, three-dimensional shapes, prism definitions).

Symmetry

Cuboids have several symmetry properties depending on the number of equal edge lengths.

have cuboids with three different edge lengths

3 bidentate axes of rotation (through the midpoints of two opposite faces),
3 mirror planes (3 planes through four edge centers each),
3 rotational reflections (by 180° with the planes through four edge centers each)

cuboids with two different edge lengths (square straight prisms) have

1 fourfold axis of rotation (through the centers of two opposite squares),

2 bidentate axes of rotation (through the midpoints of two opposite rectangles),
3 rotational reflections (by 180° with the planes through four edge centers each)

Cuboids with only one edge length, the cubes, have more symmetries (see cubes - symmetry).

Each cuboid is

point-symmetrical to the center M.

Formulas

Sizes of a cuboid with edge lengths a, b, c
Volume	$V=a\cdot b\cdot c$
Sheathing	$A_{M}=2\cdot (a+b)\cdot c$
Surface area	$A_{O}=2\cdot (a\cdot b+a\cdot c+b\cdot c)$
Circumferential radius	$r_{u}={\tfrac {d}{2}}={\tfrac {1}{2}}\cdot {\sqrt {a^{2}+b^{2}+c^{2}}}$
Space diagonal	$d=2\cdot r_{u}={\sqrt {a^{2}+b^{2}+c^{2}}}$
Area diagonals	$d_{a}={\sqrt {b^{2}+c^{2}}}$
	$d_{b}={\sqrt {c^{2}+a^{2}}}$
	$d_{c}={\sqrt {a^{2}+b^{2}}}$
Volume to sphere volume ratio	${\frac {V}{V_{UK}}}={\frac {6\cdot a\cdot b\cdot c}{\pi \cdot (a^{2}+b^{2}+c^{2})^{\frac {3}{2}}}}$
Solid angle in the corners	$\Omega ={\frac {\pi }{2}}\;\mathrm {sr} \;\approx 1{,}5708\;\mathrm {sr}$

Optimization problems and the cube

There are several optimization problems for cuboids. If one is looking for a cuboid which is

given length of the diagonal or given sphere volume the maximum surface area
given length of the diagonal or given sphere volume the maximum volume
given surface area the minimum length of the diagonal or the minimum volume of the sphere
given surface area the maximum volume
given volume the minimum length of the diagonal or the minimum sphere volume
given volume the minimum surface area

then the solution in each case is the cube.

In each case, two of the six optimization problems are basically the same problem with other given sizes, so that there are actually only three different optimization problems. For the optimization problems mentioned, the cube is the cuboid we are looking for. Of course, this is not true for all optimization problems.

That the optimization problems for the length of the diagonal and the circumsphere volume $V_{UK}$ each have the same solution is obvious, because the circumsphere volume $V_{UK}={\frac {4}{3}}\cdot \pi \cdot r_{u}^{3}={\frac {1}{6}}\cdot \pi \cdot d^{3}$ is a continuous and strictly monotonically increasing function with function variable

cuboid with the largest volume is sought for a given circumsphere radius, then the edge lengths , , of the cuboid using the partial derivatives of the volume function $V(a,b)=a\cdot b\cdot c=a\cdot b\cdot {\sqrt {d^{2}-a^{2}-b^{2}}}=a\cdot b\cdot {\sqrt {4\cdot r_{u}^{2}-a^{2}-b^{2}}}$ compute or with proof by contradiction:

Suppose that any cuboid with at least two different edge lengths, for example and , would have the largest volume. Its circumspherical radius is ${\frac {1}{2}}\cdot {\sqrt {a^{2}+b^{2}+c^{2}}}$ and its volume $a\cdot b\cdot c$ . Then another cuboid, namely the cuboid with edge lengths ${\frac {1}{2}}\cdot {\sqrt {2\cdot a^{2}+2\cdot b^{2}}}$ , ${\frac {1}{2}}\cdot {\sqrt {2\cdot a^{2}+2\cdot b^{2}}}$ and the same circumsphere radius ${\frac {1}{2}}\cdot {\sqrt {a^{2}+b^{2}+c^{2}}}$ and the volume ${\frac {a^{2}+b^{2}}{2}}\cdot c$ . Because of the inequality from the arithmetic and geometric mean ${\sqrt {a\cdot b}}\leq {\frac {a+b}{2}}$ , because $a\neq b$ and $c>0$ holds $a\cdot b<{\frac {a^{2}+b^{2}}{2}}$ and $a\cdot b\cdot c<{\frac {a^{2}+b^{2}}{2}}\cdot c$ .

So the arbitrary cuboid with at least two different edge lengths cuboid has a smaller volume than the other cuboid. From this follows that a cuboid with at least two different edge lengths cannot have the largest volume and finally that the cuboid with only one edge length, i.e. the cube with 12 edges of equal length, has the largest volume of all cuboids with a given radius of revolution.

Decisive for this proof by contradiction is here that the volume of the cuboid must be finite, because it is obviously smaller than the volume of the circumsphere, and that the volume function is continuous.