Rhombus


In plane Euclidean geometry, a rhombus 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, and the latter sometimes refers specifically to a rhombus with a 45° angle.
Every rhombus is simple, and is a special case of a parallelogram and a kite. A rhombus with right angles is a square.

Etymology

The word "rhombus" comes from Greek ῥόμβος, meaning something that spins, which derives from the verb ῥέμβω, meaning "to turn round and round." 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.
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.

Characterizations

A simple quadrilateral is a rhombus if and only if it is any one of the following:
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. 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 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. That is, it has an inscribed circle that is tangent to all four sides.

Diagonals

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.

Inradius

The inradius, denoted by, can be expressed in terms of the diagonals and as
or in terms of the side length and any vertex angle or as

Area

As for all parallelograms, the area K of a rhombus is the product of its base and its height. 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 :
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, which is the determinant of the two vectors' Cartesian coordinates: K = x1y2 – x2y1.

Dual properties

The dual polygon of a rhombus is a rectangle:
The sides of a rhombus centered at the origin, with diagonals each falling on an axis, consist of all points satisfying
The vertices are at and This is a special case of the superellipse, with exponent 1.

Other properties

A rhombohedron is a three-dimensional figure like a cuboid, 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 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 trapezo-rhombic dodecahedron is a convex polyhedron with 6 rhombic and 6 trapezoidal faces.
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.