Rhombus

      Rhombus 

plural: plural rhombi or rhombuses
equilateral quadrilateral

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Every rhombus is a parallelogram, and a rhombus with right angles is a square. (Euclid's original definition and some English dictionaries' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition.)

The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral.

A quadrilateral

A rhombus is actually just a special type of parallelogram. It therefore has all the properties of a parallelogram.

Its a bit like a square that can 'lean over'
Sometimes called a 'diamond' or 'lozenge' shape.

Properties of a rhombus

Area	There are several ways to find the area of a rhombus. The most common is (base × altitude). Each is described in Area of a rhombus
Diagonals	Each of the two diagonals is the perpendicular bisector of the other.

Area of a rhombus

Three different ways to calculate the area of a rhombus are given below, with a formula for each. 

1. The "base times height" method

First pick one side to be the base. Any one will do, they are all the same length. Then determine the altitude - the perpendicular distance from the chosen base to the opposite side. The area is the product of these two, or, as a formula:

where b is the length of the base a is the altitude (height).

2. The "diagonals" method

Another simple formula for the area of a rhombus when you know the lengths of the diagonals. The area is half the product of the diagonals. As a formula:

where d1 is the length of a diagonal d2 is the length of the other diagonal

3. Using trigonometry

If you are familiar with trigonometry, there is a handy formula when you know the length of a side and any angle:

where s is the length of any side a is any interior angle sin  is the sine function

It may seem odd at first that you can use any angle since they are not all equal. But the angles are either equal or supplementary, and supplementary angles have the same sine.

Dual properties

The dual polygon of a rhombus is a rectangle:^[8]

• A rhombus has all sides equal, while a rectangle has all angles equal.
• A rhombus has alternate angles equal, while a rectangle has alternate sides equal.
• A rhombus has an inscribed circle, while a rectangle has a circumcircle.
• A rhombus has an axis of symmetry through each pair of opposite vertex angles, while a rectangle has an axis of symmetry through each pair of opposite sides.
• The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length.