In plane

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, a rhombus (plural rhombi or rhombuses) is a

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the

diamonds Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, ...

suit in

playing card A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a f ...

s which resembles the projection of an octahedral

diamond Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, b ...

, or a

lozenge Lozenge or losange may refer to: *Lozenge (shape), a type of rhombus *Throat lozenge, a tablet intended to be dissolved slowly in the mouth to suppress throat ailments *Lozenge (heraldry), a diamond-shaped object that can be placed on the field of ...

, though the former sometimes refers specifically to a rhombus with a 60°

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...

(which some authors call a calisson after the French sweet – also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

(non-self-intersecting), and is a special case of a

parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...

and a kite. A rhombus with right angles is a square.

Etymology

The word "rhombus" comes from grc, ῥόμβος, rhombos, meaning something that spins, which derives from the verb , romanized: , meaning "to turn round and round." The word was used both by

Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

and

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...

, who used the term "solid rhombus" for a bicone, two right circular cones sharing a common base. The surface we refer to as ''rhombus'' today is a cross section of the bicone on a plane through the apexes of the two cones.

Characterizations

(non- self-intersecting) quadrilateral is a rhombus

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...

it is any one of the following: *a

in which a

diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...

bisects an interior angle *a parallelogram in which at least two consecutive sides are equal in length *a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram) *a quadrilateral with four sides of equal length (by definition) *a quadrilateral in which the diagonals are

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...

and bisect each other *a quadrilateral in which each diagonal bisects two opposite interior angles *a quadrilateral ''ABCD'' possessing a point ''P'' in its plane such that the four triangles ''ABP'', ''BCP'', ''CDP'', and ''DAP'' are all congruent *a quadrilateral ''ABCD'' in which the incircles in triangles ''ABC'', ''BCD'', ''CDA'' and ''DAB'' have a common point

Basic properties

Every rhombus has two

s connecting pairs of opposite vertices, and two pairs of parallel sides. Using congruent

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

s, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties: * Opposite

s of a rhombus have equal measure. * The two diagonals of a rhombus are

; that is, a rhombus is an orthodiagonal quadrilateral. * Its diagonals bisect opposite angles. The first property implies that every rhombus is a

. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel; adjacent angles are supplementary; the two diagonals bisect one another; any line through the midpoint bisects the area; and the sum of the squares of the sides equals the sum of the squares of the diagonals (the parallelogram law). Thus denoting the common side as ''a'' and the diagonals as ''p'' and ''q'', in every rhombus :

\displaystyle 4a^2=p^2+q^2.

Not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals (the second property) is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite. Every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus. A rhombus is a

tangential quadrilateral In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called ...

. That is, it has an

inscribed circle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incen ...

that is tangent to all four sides. Rhombus1

Diagonals

The length of the diagonals ''p = AC'' and ''q = BD'' can be expressed in terms of the rhombus side ''a'' and one vertex angle ''α'' as :

p=a\sqrt

and :

q=a\sqrt.

These formulas are a direct consequence of the law of cosines.

Inradius

The inradius (the radius of a circle

inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...

in the rhombus), denoted by , can be expressed in terms of the diagonals and as :

r = \frac,

or in terms of the side length and any vertex angle or as :

r = \frac = \frac.

Area

As for all

s, the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...

''K'' of a rhombus is the product of its base and its

height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is ab ...

(''h''). The base is simply any side length ''a'': :

K = a \cdot h .

The area can also be expressed as the base

squared A square is a regular quadrilateral with four equal sides and four right angles. Square or Squares may also refer to: Mathematics and science *Square (algebra), multiplying a number or expression by itself *Square (cipher), a cryptographic block ...

times the sine of any angle: :

K = a^2 \cdot \sin \alpha = a^2 \cdot \sin \beta ,

or in terms of the height and a

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

: :

K=\frac ,

or as half the product of the

s ''p'', ''q'': :

K = \frac ,

or as the

semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate ...

times the

radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...

of the

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

in the rhombus (inradius): :

K = 2a \cdot r .

Another way, in common with parallelograms, is to consider two adjacent sides as vectors, forming a bivector, so the area is the magnitude of the bivector (the magnitude of the vector product of the two vectors), which is the

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...

of the two vectors' Cartesian coordinates: ''K'' = ''x''₁''y''₂ – ''x''₂''y''₁.

Dual properties

The

dual polygon In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. Properties Regular polygons are self-dual. The dual of an isogonal (vertex-transitive) polygon is an isotoxal (ed ...

of a rhombus is a

rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram contain ...

:de Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", '' Mathematical Gazette'' 95, March 2011, 102-107. *A rhombus has all sides equal, while a rectangle has all angles equal. *A rhombus has opposite angles equal, while a rectangle has opposite sides equal. *A rhombus has an inscribed circle, while a rectangle has a circumcircle. *A rhombus has an axis of symmetry through each pair of opposite vertex angles, while a rectangle has an axis of symmetry through each pair of opposite sides. *The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. *The figure formed by joining the midpoints of the sides of a rhombus is a

, and vice versa.

Cartesian equation

The sides of a rhombus centered at the origin, with diagonals each falling on an axis, consist of all points (''x, y'') satisfying :

\left, \frac\\! + \left, \frac\\! = 1.

The vertices are at

(\pm a, 0)

and

(0, \pm b).

This is a special case of the superellipse, with exponent 1.

Other properties

*One of the five 2D lattice types is the rhombic lattice, also called centered rectangular lattice. * Identical rhombi can tile the 2D plane in three different ways, including, for the 60° rhombus, the

rhombille tiling In geometry, the rhombille tiling, also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape ...

. ** * Three-dimensional analogues of a rhombus include the

bipyramid A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does ...

and the bicone as a

surface of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...

As the faces of a polyhedron

Convex polyhedra with rhombi include the infinite set of rhombic zonohedrons, which can be seen as projective envelopes of

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...

s. * A rhombohedron (also called a rhombic hexahedron) is a three-dimensional figure like a

cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a c ...

(also called a rectangular parallelepiped), except that its 3 pairs of parallel faces are up to 3 types of rhombi instead of rectangles. * The rhombic dodecahedron is a convex polyhedron with 12 congruent rhombi as its faces. * The rhombic triacontahedron is a convex polyhedron with 30 golden rhombi (rhombi whose diagonals are in the golden ratio) as its faces. * The

great rhombic triacontahedron In geometry, the great rhombic triacontahedron is a nonconvex Isohedral figure, isohedral, isotoxal polyhedron. It is the Dual polyhedron, dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombus, rhomb ...

is a nonconvex isohedral,

isotoxal In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given ...

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...

with 30 intersecting rhombic faces. * The

rhombic hexecontahedron In geometry, a rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. It was described mathematically in 1940 by Helmut Unkelbach. It is topologically ident ...

is a

stellation In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specif ...

of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. * The

rhombic enneacontahedron In geometry, a rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is ...

is a polyhedron composed of 90 rhombic faces, with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim ones. * The rhombic icosahedron is a polyhedron composed of 20 rhombic faces, of which three, four, or five meet at each vertex. It has 10 faces on the polar axis with 10 faces following the equator.

References

External links

Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)
with interactive applet.

- shows three different ways to compute the area of a rhombus, with interactive applet {{Authority control Types of quadrilaterals Elementary shapes