cuboid

A cuboid, or rectangular prism, is a three-dimensional solid consisting of six rectangular faces. It is the general form of a prism with rectangular bases; when the three edge lengths are equal, the cuboid becomes a cube.

Opposite faces of a cuboid are parallel and congruent, and the edges meeting at a vertex are perpendicular to each other in a right cuboid.

The solid has 12 edges and 8 vertices. The arrangement satisfies V − E + F = 2, a consequence of Euler’s formula for convex polyhedra.

Volume is the product of its edge lengths: V = l × w × h. Surface area equals the sum of the areas of all faces: SA = 2(lw + wh + hl).

A space diagonal connects opposite vertices, with length d = sqrt(l^2 + w^2 + h^2). Each face has a diagonal, for example sqrt(l^2 + w^2) on the l-by-w face.

In coordinate form, a cuboid can be defined by the Cartesian product [0, l] × [0, w] × [0, h], with vertices at (0,0,0), (l,0,0), (0,w,0), (0,0,h), and the remaining points obtained by combining coordinates.

Applications include packaging, construction, and design, where cuboids model blocks, rooms, and containers due to their regular, straight-edged geometry. The cube is a notable special case of a cuboid with all edges equal.