Rhombus | |
---|---|

Type | quadrilateral, trapezoid, parallelogram, kite |

Edges and vertices | 4 |

Schläfli symbol | { } + { } {2 _{α}} |

Coxeter–Dynkin diagrams | |

Symmetry group | Dihedral (D_{2}), [2], (*22), order 4 |

Area | (half the product of the diagonals) |

Properties | convex, isotoxal |

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.

- Etymology
- Characterizations
- Basic properties
- Diagonals
- Inradius
- Area
- Dual properties
- Cartesian equation
- Other properties
- As the faces of a polyhedron
- See also
- References
- External links

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] }

- a parallelogram in which a diagonal 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 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^{ [9] } - a quadrilateral
*ABCD*in which the incircles in triangles*ABC*,*BCD*,*CDA*and*DAB*have a common point^{ [10] }

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:

- Opposite angles of a rhombus have equal measure.
- The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral.
- Its diagonals bisect opposite angles.

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.

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

and

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* = *x*_{1}*y*_{2} – *x*_{2}*y*_{1}.^{ [12] }

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

- 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 rectangle, and vice versa.

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.

- 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.

As topological square tilings | As 30-60 degree rhombille tiling | |
---|---|---|

- Three-dimensional analogues of a rhombus include the bipyramid and the bicone as a surface of revolution.

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

- A rhombohedron (also called a rhombic hexahedron) is a three-dimensional figure like a cuboid (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 is a nonconvex isohedral, isotoxal polyhedron with 30 intersecting rhombic faces.
- The rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry.
- The rhombic enneacontahedron 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.

Isohedral | Isohedral golden rhombi | 2-isohedral | 3-isohedral | ||
---|---|---|---|---|---|

Trigonal trapezohedron | Rhombic dodecahedron | Rhombic triacontahedron | Rhombic icosahedron | Rhombic enneacontahedron | Rhombohedron |

- Merkel-Raute
- Rhombus of Michaelis, in human anatomy
- Rhomboid, either a parallelepiped or a parallelogram that is neither a rhombus nor a rectangle
- Rhombic antenna
- Rhombic Chess
- Flag of the Department of North Santander of Colombia, containing four stars in the shape of a rhombus
- Superellipse (includes a rhombus with rounded corners)

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

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 .

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.

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*.

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.

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.

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*.

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.

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**.

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.

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.

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*.

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.

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.

**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.

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.

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

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.

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.

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.

- ↑ Alsina, Claudi; Nelsen, Roger B. (31 December 2015).
*A Mathematical Space Odyssey: Solid Geometry in the 21st Century*. ISBN 9781614442165. - ↑ Note: Euclid's original definition and some English dictionaries' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition.
- ↑ Weisstein, Eric W. "Square".
*MathWorld*. inclusive usage - ↑ ῥόμβος Archived 2013-11-08 at the Wayback Machine , Henry George Liddell, Robert Scott,
*A Greek-English Lexicon*, on Perseus - ↑ ρέμβω Archived 2013-11-08 at the Wayback Machine , Henry George Liddell, Robert Scott,
*A Greek-English Lexicon*, on Perseus - ↑ "The Origin of Rhombus". Archived from the original on 2015-04-02. Retrieved 2005-01-25.
- ↑ 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.
- ↑ 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. - ↑ Paris Pamfilos (2016), "A Characterization of the Rhombus",
*Forum Geometricorum***16**, pp. 331–336, Archived 2016-10-23 at the Wayback Machine - ↑ "IMOmath, "26-th Brazilian Mathematical Olympiad 2004"" (PDF). Archived (PDF) from the original on 2016-10-18. Retrieved 2020-01-06.
- 1 2 Weisstein, Eric W. "Rhombus".
*MathWorld*. - ↑ WildLinAlg episode 4 Archived 2017-02-05 at the Wayback Machine , Norman J Wildberger, Univ. of New South Wales, 2010, lecture via youtube
- ↑ de Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons",
*Mathematical Gazette*95, March 2011, 102-107.

Look up ** rhombus ** in Wiktionary, the free dictionary.

Wikimedia Commons has media related to Rhombi .

- Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)
- Rhombus definition, Math Open Reference with interactive applet.
- Rhombus area, Math Open Reference - shows three different ways to compute the area of a rhombus, with interactive applet

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