The namerhombus comes fromGreekῥόμβοςrhómbos, meaning something that spins, such as abullroarer or an ancient precursor of thebutton whirligig.[2] The word was used both byEuclid andArchimedes, who also used the term "solid rhombus" for abicone, two right circularcones sharing a common base.[3] A planarrhombus is across section of a bicone.
The namediamond comes from the shape of anoctahedraldiamond gemstone; thediamonds suit inplaying cards is named after the shape – it was originally calledcarreaux (lit. "squares") in French.[4] In the context ofpolyiamonds, shapes likepolyominos but constructed from equilateral triangles, adiamond is a rhombus with a 60° angle.
The etymology oflozenge is uncertain. It might come from a shape of somelauzinaj almond pastries, or from the shape of tombstones. A lozenge is often specifically to a rhombus with a 45° angle.
Acalisson is a type of rhombus-shaped French sweet.[5]
a parallelogram in which at least two consecutive sides are equal in length
a parallelogram in which the diagonals are perpendicular (anorthodiagonal parallelogram)
a quadrilateral with four sides of equal length (by definition)
a quadrilateral in which the diagonals areperpendicular andbisect each other
a quadrilateral in which each diagonal bisects two opposite interior angles
a quadrilateralABCD possessing a pointP in its plane such that the four trianglesABP,BCP,CDP, andDAP are allcongruent[8]
a quadrilateralABCD in which theincircles in trianglesABC,BCD,CDA andDAB have a common point[9]
Basic properties
Every rhombus has twodiagonals connecting pairs of opposite vertices, and two pairs of parallel sides. Usingcongruenttriangles, one canprove that the rhombus issymmetric across each of these diagonals. It follows that any rhombus has the following properties:
The first property implies that every rhombus is aparallelogram. A rhombus therefore has all of theproperties of a parallelogram: for example, opposite sides are parallel; adjacent angles aresupplementary; the two diagonalsbisect 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 (theparallelogram law). Thus denoting the common side asa and the diagonals asp andq, 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 akite. Every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus.
A rhombus. Each angle marked with a black dot is a right angle. The heighth is the perpendicular distance between any two non-adjacent sides, which equals the diameter of the circle inscribed. The diagonals of lengthsp andq are the red dotted line segments.
Diagonals
The length of the diagonalsp = AC andq = BD can be expressed in terms of the rhombus sidea and one vertex angleα as
and
These formulas are a direct consequence of thelaw of cosines.
Inradius
The inradius (the radius of a circleinscribed in the rhombus), denoted byr, can be expressed in terms of the diagonalsp andq as[10]
or in terms of the side lengtha and any vertex angleα orβ as
Area
As for allparallelograms, theareaK of a rhombus is the product of itsbase and itsheight (h). The base is simply any side lengtha:
The area can also be expressed as thebasesquared times the sine of any angle:
Another way, in common with parallelograms, is to consider two adjacent sides as vectors, forming abivector, so the area is the magnitude of the bivector (the magnitude of the vector product of the two vectors), which is thedeterminant of the two vectors' Cartesian coordinates:K =x1y2 –x2y1.
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 acircumcircle.
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 arectangle, 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
The vertices are at and This is a special case of thesuperellipse, with exponent 1.
Convex polyhedra with rhombi include the infinite set of rhombiczonohedrons, which can be seen as projective envelopes ofhypercubes.
Arhombohedron (also called a rhombic hexahedron) is a three-dimensional figure like acuboid (also called a rectangular parallelepiped), except that its 3 pairs of parallel faces are up to 3 types of rhombi instead of rectangles.
Therhombic 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.
Therhombic 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.
Superellipse (includes a rhombus with rounded corners)
Notes
^Euclid's original definition and some English dictionaries' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition.[1]
^ῥόμβοςArchived 2013-11-08 at theWayback Machine, Henry George Liddell, Robert Scott,A Greek-English Lexicon, on Perseus.Hoorn, Gerard van (1951).Choes and Anthesteria. Brill Archive. Retrieved22 August 2022. Also see:ρέμβωArchived 2013-11-08 at theWayback Machine, Henry George Liddell, Robert Scott,A Greek-English Lexicon, on Perseus
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