Parallelepiped

In geometry, a parallelepiped is a special three-dimensional shape made up of six parallelograms. You can think of it like a stretched box where each face is a slanted rectangle. It is similar to how a cube relates to a square in flat shapes.

There are a few ways to describe a parallelepiped. It can be a shape with six sides where each pair of opposite sides are parallel. Another way to see it is as a solid with six faces, and every face is a parallelogram. You can also imagine it as a prism that starts with a parallelogram as its base.

Some common shapes that are types of parallelepipeds include the rectangular cuboid, which has six flat rectangular faces, the cube with six perfect square faces, and the rhombohedron made of six diamond-shaped faces called rhombus. The word "parallelepiped" comes from Ancient Greek and means a body with parallel planes.

Parallelepiped

Type	Prism Plesiohedron
Faces	6 parallelograms
Edges	12
Vertices	8
Symmetry group	C_i, [2⁺,2⁺], (×), order 2
Properties	convex, zonohedron

Properties

A parallelepiped is a 3D shape with six faces, and each face is a flat shape called a parallelogram. You can pick any pair of opposite faces to be the base of the shape, and it will still look the same. The edges of a parallelepiped come in three groups of four, and each group has edges that are all the same length.

Parallelepipeds can be made by changing the shape of a cube using straight-line math rules. Because of this, they fit together perfectly to fill space without any gaps. Each face looks like a mirror image of the face across from it.

Volume

A parallelepiped is a prism with a parallelogram as its base. The volume of a parallelepiped is found by multiplying the area of the base by its height.

The volume can also be calculated using vectors and angles between them, but the simplest way is to use the base area and height.

The volume of a tetrahedron that shares three edges of a parallelepiped is one sixth of the parallelepiped's volume.

V = | det [ a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 ] | . {\displaystyle V=\left|\det {\begin{bmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{bmatrix}}\right|.}

V = a b c 1 + 2 cos ⁡ ( α ) cos ⁡ ( β ) cos ⁡ ( γ ) − cos 2 ⁡ ( α ) − cos 2 ⁡ ( β ) − cos 2 ⁡ ( γ ) , {\displaystyle V=abc{\sqrt {1+2\cos(\alpha )\cos(\beta )\cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )}},}

Surface area

The surface area of a parallelepiped is the total area of all its sides. You can find it by adding up the areas of the six parallelogram shapes that make up the figure. This helps us understand how much space the outside of the shape covers.

Special cases by symmetry

The parallelepiped with O_h symmetry is known as a cube, which has six identical square faces.
The parallelepiped with D_4h symmetry is known as a square cuboid, which has two square faces and four matching rectangular faces.
The parallelepiped with D_3d symmetry is known as a trigonal trapezohedron, which has six matching rhombic faces.
For parallelepipeds with D_2h symmetry, there are two cases:
- Rectangular cuboid: it has six rectangular faces.
- Right rhombic prism: it has two rhombic faces and four matching rectangular faces.
For parallelepipeds with C_2h symmetry, there are two cases:
- Right parallelogrammic prism: it has four rectangular faces and two parallelogrammic faces.
- Oblique rhombic prism: it has two rhombic faces, with the other faces arranged in pairs that are mirror images of each other.

Octahedral symmetry subgroup relations with inversion center

Special cases of the parallelepiped

Form	Cube	Square cuboid	Trigonal trapezohedron	Rectangular cuboid	Right rhombic prism	Right parallelogrammic prism	Oblique rhombic prism
Constraints	a = b = c {\displaystyle a=b=c} α = β = γ = 90 ∘ {\displaystyle \alpha =\beta =\gamma =90^{\circ }}	a = b {\displaystyle a=b} α = β = γ = 90 ∘ {\displaystyle \alpha =\beta =\gamma =90^{\circ }}	a = b = c {\displaystyle a=b=c} α = β = γ {\displaystyle \alpha =\beta =\gamma }	α = β = γ = 90 ∘ {\displaystyle \alpha =\beta =\gamma =90^{\circ }}	a = b {\displaystyle a=b} α = β = 90 ∘ {\displaystyle \alpha =\beta =90^{\circ }}	α = β = 90 ∘ {\displaystyle \alpha =\beta =90^{\circ }}	a = b {\displaystyle a=b} α = β {\displaystyle \alpha =\beta }
Symmetry	O_h order 48	D_4h order 16	D_3d order 12	D_2h order 8		C_2h order 4
Image
Faces	6 squares	2 squares, 4 rectangles	6 rhombi	6 rectangles	4 rectangles, 2 rhombi	4 rectangles, 2 parallelograms	2 rhombi, 4 parallelograms

Perfect parallelepiped

A perfect parallelepiped is a special shape where all the edges, face diagonals, and space diagonals have whole number lengths. In 2009, many of these special shapes were found, solving a long-standing question asked by Richard Guy. One example has edges of 271, 106, and 103 units.

Some of these special shapes have two rectangle-like faces, but we still do not know if any exist with all faces rectangle-like, which would be called a perfect cuboid.

Parallelotope

Coxeter used the term parallelotope to describe shapes that are like parallelepipeds but in higher dimensions. Today, we often use "parallelepiped" to talk about these shapes in any number of dimensions.

In n-dimensional space, such a shape is called an n-dimensional parallelotope, or simply an n-parallelotope. For example, a parallelogram is a 2-parallelotope, and a parallelepiped is a 3-parallelotope.

The diagonals of an n-parallelotope all meet at one point and are cut in half by that point. Inversion around this point leaves the shape unchanged. The edges from one vertex form a k-frame, and the parallelotope can be built from these edges using certain math rules.

The size, or volume, of an n-parallelotope can be found using special math tools. When the space has the same number of dimensions as the parallelotope, the volume is linked to a matrix made from the edges.

Etymology

The word parallelepiped comes from Ancient Greek. It means "body with parallel plane surfaces." It is made from two parts: parallēl (parallel) and epípedon (plane surface).

The word has been used in English since 1570. Over time, its spelling has changed a little. Some older forms include parallelipipedon and parallelepipedum. Today, we usually say parallelepiped.