Last updated

A rhombus in two different orientations
Type quadrilateral, trapezoid, parallelogram, kite
Edges and vertices 4
Schläfli symbol { } + { }
Coxeter–Dynkin diagrams CDel node 1.pngCDel sum.pngCDel node 1.png
Symmetry group Dihedral (D2), [2], (*22), order 4
Area (half the product of the diagonals)
Properties convex, isotoxal
The rhombus has a square as a special case, and is a special case of a kite and parallelogram. Symmetries of square.svg
The rhombus has a square as a special case, and is a special case of a kite and parallelogram.

In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral 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 suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after the French sweet [1] also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.


Every rhombus is simple (non-self-intersecting), and is a special case of a parallelogram and a kite. A rhombus with right angles is a square. [2] [3]


The word "rhombus" comes from Ancient Greek : ῥόμβος, romanized: rhombos, meaning something that spins, [4] which derives from the verb ῥέμβω , romanized:rhémbō, meaning "to turn round and round." [5] The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for a bicone, two right circular cones sharing a common base. [6]

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.


A simple (non-self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following: [7] [8]

Basic properties

Every rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties:

The first property implies that every rhombus is a parallelogram. 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

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. [11] That is, it has an inscribed circle that is tangent to all four sides.

A rhombus. Each angle marked with a black dot is a right angle. The height h is the perpendicular distance between any two non-adjacent sides, which equals the diameter of the circle inscribed. The diagonals of lengths p and q are the red dotted line segments. Rhombus1.svg
A rhombus. Each angle marked with a black dot is a right angle. The height h is the perpendicular distance between any two non-adjacent sides, which equals the diameter of the circle inscribed. The diagonals of lengths p and q are the red dotted line segments.


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


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


The inradius (the radius of a circle inscribed in the rhombus), denoted by r, can be expressed in terms of the diagonals p and q as [11]

or in terms of the side length a and any vertex angle α or β as


As for all parallelograms, the area K of a rhombus is the product of its base and its height (h). The base is simply any side length a:

The area can also be expressed as the base squared times the sine of any angle:

or in terms of the height and a vertex angle:

or as half the product of the diagonals p, q:

or as the semiperimeter times the radius of the circle inscribed in the rhombus (inradius):

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 of the two vectors' Cartesian coordinates: K = x1y2x2y1. [12]

Dual properties

The dual polygon of a rhombus is a rectangle: [13]

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

The vertices are at and This is a special case of the superellipse, with exponent 1.

Other properties

As topological square tilings As 30-60 degree rhombille tiling
Isohedral tiling p4-55.png Isohedral tiling p4-51c.png Rhombic star tiling.png

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 hypercubes.

Example polyhedra with all rhombic faces
IsohedralIsohedral golden rhombi2-isohedral3-isohedral
TrigonalTrapezohedron.svg Rhombicdodecahedron.jpg Rhombictriacontahedron.jpg Rhombic icosahedron.png Rhombic enneacontahedron.png Rhombohedron.svg
Trigonal trapezohedron Rhombic dodecahedron Rhombic triacontahedron Rhombic icosahedron Rhombic enneacontahedron Rhombohedron

See also

Related Research Articles

<span class="mw-page-title-main">Parallelepiped</span> Hexahedron with parallelogram faces

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—parallelepiped and cube in three dimensions, parallelogram and square in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only parallelograms and parallelepipeds exist. Three equivalent definitions of parallelepiped are

<span class="mw-page-title-main">Quadrilateral</span> Polygon with four sides and four corners

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, derived from greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .

<span class="mw-page-title-main">Tetrahedron</span> Polyhedron with 4 faces

In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.

<span class="mw-page-title-main">Rectangle</span> Quadrilateral with four right angles

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 ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as  ABCD.

<span class="mw-page-title-main">Kite (geometry)</span> Quadrilateral symmetric across a diagonal

In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.

<span class="mw-page-title-main">Parallelogram</span> Quadrilateral with two pairs of parallel sides

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 equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

<span class="mw-page-title-main">Bisection</span> Division of something into two equal or congruent parts

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector.

<span class="mw-page-title-main">Cyclic quadrilateral</span> Quadrilateral whose vertices can all fall on a single circle

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

<span class="mw-page-title-main">Trapezoid</span> Convex quadrilateral with at least one pair of parallel sides

A quadrilateral with at least one pair of parallel sides is called a trapezoid in American and Canadian English. In British and other forms of English, it is called a trapezium.

<span class="mw-page-title-main">Thales's theorem</span> Angle formed by a point on a circle and the 2 ends of a diameter is a right angle

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.

<span class="mw-page-title-main">Isosceles trapezoid</span> Trapezoid symmetrical about an axis

In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of equal measure. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides are parallel, and the two other sides are of equal length. The diagonals are also of equal length. The base angles of an isosceles trapezoid are equal in measure.

<span class="mw-page-title-main">Square</span> Regular quadrilateral

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ABCD.

<span class="mw-page-title-main">Rhombic triacontahedron</span> Catalan solid with 30 faces

In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

<span class="mw-page-title-main">Tangential quadrilateral</span> Polygon whose four sides all touch a circle

In Euclidean geometry, a tangential 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 the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons.

<span class="mw-page-title-main">Varignon's theorem</span> The midpoints of the sides of an arbitrary quadrilateral form a parallelogram

Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, whose proof was published posthumously in 1731.

<span class="mw-page-title-main">Medial rhombic triacontahedron</span>

In geometry, the medial rhombic triacontahedron is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron.

<span class="mw-page-title-main">Golden rhombus</span> Rhombus with diagonals in the golden ratio

In geometry, a golden rhombus is a rhombus whose diagonals are in the golden ratio:

<span class="mw-page-title-main">Orthodiagonal quadrilateral</span>

In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

<span class="mw-page-title-main">Ex-tangential quadrilateral</span> Convex 4-sided polygon whose sidelines are all tangent to an outside circle

In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. It has also been called an exscriptible quadrilateral. The circle is called its excircle, its radius the exradius and its center the excenter. The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The ex-tangential quadrilateral is closely related to the tangential quadrilateral.

<span class="mw-page-title-main">Equidiagonal quadrilateral</span>

In Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified first according to whether they were equidiagonal and then into more specialized types.


  1. Alsina, Claudi; Nelsen, Roger B. (31 December 2015). A Mathematical Space Odyssey: Solid Geometry in the 21st Century. ISBN   9781614442165.
  2. Note: Euclid's original definition and some English dictionaries' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition.
  3. Weisstein, Eric W. "Square". MathWorld . inclusive usage
  4. ῥόμβος Archived 2013-11-08 at the Wayback Machine , Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  5. ρέμβω Archived 2013-11-08 at the Wayback Machine , Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  6. "The Origin of Rhombus". Archived from the original on 2015-04-02. Retrieved 2005-01-25.
  7. Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition Archived 2020-02-26 at the Wayback Machine ", Information Age Publishing, 2008, pp. 55-56.
  8. Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry Archived 2019-09-01 at the Wayback Machine , Mathematical Association of America, 2010, p. 53.
  9. Paris Pamfilos (2016), "A Characterization of the Rhombus", Forum Geometricorum 16, pp. 331–336, Archived 2016-10-23 at the Wayback Machine
  10. "IMOmath, "26-th Brazilian Mathematical Olympiad 2004"" (PDF). Archived (PDF) from the original on 2016-10-18. Retrieved 2020-01-06.
  11. 1 2 Weisstein, Eric W. "Rhombus". MathWorld .
  12. WildLinAlg episode 4 Archived 2017-02-05 at the Wayback Machine , Norman J Wildberger, Univ. of New South Wales, 2010, lecture via youtube
  13. de Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", Mathematical Gazette 95, March 2011, 102-107.