Rhombus

 In plane Euclidean geometry, a rhombus (plural 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.

Rhombus
A rhombus in two different orientations
Type	quadrilateral, trapezoid, parallelogram, kite
Edges and vertices	4
Schläfli symbol	{ } + { } {2α}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D2), [2], (*22), order 4
Area	${\displaystyle K={\frac {p\cdot q}{2}}}$ (half the product of the diagonals)
Properties	convex, isotoxal

The rhombus has a square as a special case, and is a special case of a kite and parallelogram.

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][3]

Basic properties

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:

• Opposite angles of a rhombus have equal measure.
• The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral.
• Its diagonals bisect opposite angles.

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

${\displaystyle \displaystyle 4a^{2}=p^{2}+q^{2}.}$

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.^[11] That is, it has an inscribed circle that is tangent to all four sides.

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.