Rhombus
In plane Euclidean geometry, a rhombus (pl.: rhombi or rhombuses) 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 (which some authors call a calisson after the French sweet[1]—also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.
For other uses, see Rhombus (disambiguation).Rhombus
Every rhombus is simple (non-self-intersecting), and is a special case of a parallelogram and a kite. A rhombus with right angles is a square.[2]
Etymology
The word "rhombus" comes from Ancient Greek: ῥόμβος, romanized: rhómbos, meaning something that spins,[3] which derives from the verb ῥέμβω, romanized: rhémbō, meaning "to turn round and round."[4] 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.[5]
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.
A simple (non-self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:[6][7]
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
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.[10] That is, it has an inscribed circle that is tangent to all four sides.
The dual polygon of a rhombus is a rectangle:[12]