Rhombus

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For other uses, see Rhombus (disambiguation).
rhombus
Rhombus.svg
Two rhombi.
Type quadrilateral, bipyramid
Edges and vertices 4
Schläfli symbol { } + { } or 2{ }
Coxeter diagram CDel node f1.pngCDel 2.pngCDel node f1.png
Symmetry group Dih2, [2], (*22), order 4
Area \tfrac{pq}{2} (half the product of the diagonals)
Dual polygon rectangle
Properties convex, isotoxal

In Euclidean geometry, a rhombus (◊), plural rhombi or rhombuses, is a simple (non-self-intersecting) quadrilateral all of whose four sides 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, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.

Every rhombus is a parallelogram, and a rhombus with right angles is a square.[1][2]

Etymology[edit]

The word "rhombus" comes from Greek ῥόμβος(rhombos), meaning something that spins,[3] which derives from the verb ρέμβω (rhembō), meaning "to turn round and round".[4] The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for two right circular cones sharing a common base.[5]

Characterizations[edit]

A simple (non self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:[6][7]

  • 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 parallelogram in which at least two consecutive sides are equal in length
  • a parallelogram in which the diagonals are perpendicular
  • a parallelogram in which a diagonal bisects an interior angle

Basic properties[edit]

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

\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.[8] That is, it has an inscribed circle that is tangent to all four sides.

Area[edit]

The height h is the perpendicular distance between any two non-adjacent sides, or the diameter of the circle inscribed.

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

A = a \cdot h .

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

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

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

A = \frac{p \cdot q}{2} ,

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

A = 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 of the two vectors' Cartesian coordinates: area = x1 × y2 - x2 × y1.[9]

Inradius[edit]

The inradius (the radius of the incircle), denoted by r, can be expressed in terms of the diagonals p and q as[8]

r = \frac{p \cdot q}{2\sqrt{p^2+q^2}}.

Dual properties[edit]

The dual polygon of a rhombus is a rectangle:[10]

  • 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, in order, the midpoints of the sides of a rhombus is a rectangle and vice-versa.

Other properties[edit]

As topological square tilings As 30-60 degree rhombille tiling
Isohedral tiling p4-55.png Isohedral tiling p4-51c.png Rhombic star tiling.png
Some polyhedra with all rhombic faces
Identical rhombi Two types of rhombi
Rhombohedron.svg Rhombicdodecahedron.jpg Rhombictriacontahedron.jpg Rhombic icosahedron.png Rhombic enneacontahedron.png
Rhombohedron Rhombic dodecahedron Rhombic triacontahedron Rhombic icosahedron Rhombic enneacontahedron

As a Varignon parallelogram[edit]

The Varignon parallelogram of an equidiagonal quadrilateral is a rhombus.[11]

See also[edit]

References[edit]

  1. ^ Note: Euclid's original definition and some English dictionaries' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition.
  2. ^ Weisstein, Eric W., "Square", MathWorld. inclusive usage
  3. ^ ῥόμβος, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  4. ^ ρέμβω, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  5. ^ The Origin of Rhombus
  6. ^ Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, pp. 55-56.
  7. ^ Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry, Mathematical Association of America, 2010, p. 53.
  8. ^ a b Weisstein, Eric W., "Rhombus", MathWorld.
  9. ^ WildLinAlg episode 4, Norman J Wildberger, Univ. of New South Wales, 2010, lecture via youtube
  10. ^ de Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", Mathematical Gazette 95, March 2011, 102-107.
  11. ^ de Villiers, Michael (2009), Some Adventures in Euclidean Geometry, Dynamic Mathematics Learning, p. 58, ISBN 9780557102952 .

External links[edit]