Rhombus

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rhombus
Rhombus.svg
Two rhombi.
Typequadrilateral, bipyramid
Edges and vertices4
Schläfli symbol{ } + { } or 2{ }
Coxeter diagramCDel node f1.pngCDel 2.pngCDel node f1.png
Symmetry groupDih2, [2], (*22), order 4
Area\tfrac{pq}{2}
Dual polygonrectangle
Propertiesconvex, isotoxal
 
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rhombus
Rhombus.svg
Two rhombi.
Typequadrilateral, bipyramid
Edges and vertices4
Schläfli symbol{ } + { } or 2{ }
Coxeter diagramCDel node f1.pngCDel 2.pngCDel node f1.png
Symmetry groupDih2, [2], (*22), order 4
Area\tfrac{pq}{2}
Dual polygonrectangle
Propertiesconvex, isotoxal

In Euclidean geometry, a rhombus (◊), plural rhombi or rhombuses, is a simple (non-self-intersecting) 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, 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]

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 of its sides.

Area[edit]

Rhombus1.svg

As for all parallelograms, the area A of a rhombus is the product of its base and its height. The base is simply any side length a, and the height h is the perpendicular distance between any two non-adjacent sides:

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) 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]

Other properties[edit]

As topological square tilingsAs 30-60 degree rhombille tiling
Isohedral tiling p4-55.pngIsohedral tiling p4-51c.pngRhombic star tiling.png
Some polyhedra with all rhombic faces
Identical rhombiTwo types of rhombi
Rhombohedron.svgRhombicdodecahedron.jpgRhombictriacontahedron.jpgRhombic icosahedron.pngRhombic enneacontahedron.png
RhombohedronRhombic dodecahedronRhombic triacontahedronRhombic icosahedronRhombic enneacontahedron

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.

External links[edit]