The rhombus has a square as a special case, and is a special case of a kite and parallelogram.
In plane 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 which resembles the projection of an octahedraldiamond, 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.
The word "rhombus" comes from Greekῥόμβος (rhombos), meaning something that spins, which derives from the verb ῥέμβω (rhembō), meaning "to turn round and round." The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for two right circular cones sharing a common base.
The surface we refer to as rhombus today is a cross section of this solid rhombus through the apex of each of the two cones.
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
a quadrilateral ABCD in which the incircles in triangles ABC, BCD, CDA and DAB have a common point
Every rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides. Using congruenttriangles, 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
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. Each angle marked with a black dot is a right angle. The height h is the perpendicular distance between any two non-adjacent sides, which equals the diameter of the circle inscribed. The diagonals of lengths p and q are the red dotted line segments.
As for all parallelograms, the areaK 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:
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 = x1y2 – x2y1.
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