Rhombus

In geometry Geometry

Geometry arose as the field of knowledge dealing with spatial relationships....

, a rhombus is a quadrilateral Quadrilateral

In geometry [i], a quadrilateral is a polygon [i] with four sides and four vertices. ...

in which all of the sides are of equal length, i.e., it is an equilateral Equilateral

In geometry [i], an equilateral polygon [i] has all sides of the same length. ...

quadrangle. In colloquial usage the shape is often described as a diamond or lozenge. In any rhombus, opposite sides will be parallel. Thus, the rhombus is a special case of the parallelogram Parallelogram

A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides....

. One suggestive analogy is that the rhombus is to the parallelogram as the square is to the rectangle Rectangle

In geometry [i], a rectangle is defined as a quadrilateral [i] where all four of its angles are right angle [i] ...

. If any angle of a rhombus is a right angle, then all the angles of that rhombus are right angles, and it is then a rectangle and a square.

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Encyclopedia

In geometry Geometry

Geometry arose as the field of knowledge dealing with spatial relationships....

, a rhombus is a quadrilateral Quadrilateral

In geometry [i], a quadrilateral is a polygon [i] with four sides and four vertices. ...

in which all of the sides are of equal length, i.e., it is an equilateral Equilateral

In geometry [i], an equilateral polygon [i] has all sides of the same length.
...

quadrangle. In colloquial usage the shape is often described as a diamond or lozenge.

In any rhombus, opposite sides will be parallel. Thus, the rhombus is a special case of the parallelogram Parallelogram

A kite is a flying tethered man-made object....

, that is, a quadrilateral with two pairs of equal adjacent sides. The opposite sides of a kite are not parallel unless the kite is also a rhombus.

The rhombus has the same symmetry Symmetry

Symmetry is a characteristic feature of geometrical [i] shapes, system [i]s, equation [i]s, and ...

as the rectangle and is its dual: the vertices of one correspond to the sides of the other.

A rhombus in the plane has five degrees of freedom: one for the shape, one for the size, one for the orientation, and two for the position.

The diagonals of a rhombus are perpendicular Perpendicular

In geometry [i], two lines [i] are considered perpendicular if one falls on the other in such a way ...

to each other. Hence, by joining the midpoints of each side, a rectangle Rectangle

In geometry [i], a rectangle is defined as a quadrilateral [i] where all four of its angles are right angle [i] ...

can be produced.

One of the five 2D lattice types is the rhombic lattice, also called centered rectangular lattice.

Consecutive angles of a rhombus are supplementary.

Proof

The diagonals are perpendicular.

Let A, B, C and D be the vertices of the rhombus, named in agreement with the figure . Using to represent the vector from A to B, one notices that

.

The last equality comes from the parallelism of CD and AB.
Taking the inner product Inner product space

In mathematics [i], an inner product space is a vector space [i] with additional structure, an inner...

,

since the norms of AB and BC are equal and since the inner product is bilinear and symmetric. The inner product of the diagonals is zero if and only if they are perpendicular.

Area

The area Area

Area is a physical quantity [i] expressing the size of a part of a surface [i]. ...

of any rhombus is one half the product of the lengths of its diagonals:

Because the rhombus is a parallelogram Parallelogram

A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides....

with four equal sides, the area also equals the length of a side multiplied by the perpendicular distance between two opposite sides:

Origin

The origin of the word rhombus is from the Greek word for something that spins. Euclid Euclid

Euclid , a Greek [i] mathematician [i], who lived in Alexandria [i], Hellenistic Egypt [i], alm ...

uses the word ??��??; and in his translation Heath says it is apparently drawn from the Greek word ?e��?, to turn round and round. He also points out that Archimedes Archimedes

Archimedes was an ancient Greek [i] mathematician [i], physicist [i], engineer [i], astronomer [i] ...

used the term solid rhombus for two right circular cone Cone

Cone [i] is a basic geometrical shape. ...

s sharing a common base. For more on the origin of the word, see rhombus at the .

External links

With interactive applet.
Shows three different ways to compute the area of a rhombus, with interactive applet.