Rhombus

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For other uses of the word rhombus, see Rhombus (disambiguation)

This shape is a rhombus

In geometry, a rhombus (or rhomb; plural rhombi) is a quadrilateral in which all of the sides are of equal length, i.e., it is an equilateral quadrangle. If any angle of an equilateral quadrangle is a right angle, then all its angles are right angles and it is also a square. 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. One suggestive analogy is that the rhombus is to the parallelogram as the square is to the rectangle. A rhombus is also a special case of a kite, 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.

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 to each other. Hence, by joining the midpoints of each side, a rectangle can be produced.

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

Adjacent angles of a rhombus are supplementary.

1 Proof that the diagonals are perpendicular
2 Area
3 Origin
4 External links

[edit] Proof that the diagonals are perpendicular

If A, B, C and D were the vertices of the rhombus, named in agreement with the figure (higher on this page). Using $\overrightarrow{AB}$ to represent the vector from A to B, one notices that
$\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}$
$\overrightarrow{BD} = \overrightarrow{BC}+ \overrightarrow{CD}= \overrightarrow{BC}- \overrightarrow{AB}$ .
The last equality comes from the parallelism of CD and AB. Taking the inner product,

$<\overrightarrow{AC}, \overrightarrow{BD}> = <\overrightarrow{AB} + \overrightarrow{BC}, \overrightarrow{BC} - \overrightarrow{AB}>$

$= <\overrightarrow{AB}, \overrightarrow{BC}> - <\overrightarrow{AB}, \overrightarrow{AB}> + <\overrightarrow{BC}, \overrightarrow{BC}> - <\overrightarrow{BC}, \overrightarrow{AB}>$

= 0

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.

[edit] Area

The area of any rhombus is one half the product of the lengths of its diagonals:
$A=\frac{D_1 \times D_2}{2}$
Because the rhombus is a parallelogram with four equal sides, the area also equals the length of a side (B) multiplied by the perpendicular distance between two opposite sides(H):
$A=B \times H$

[edit] Origin

The origin of the word rhombus is from the Greek word for something that spins. Euclid uses the word ρομβος; and in his translation Heath says it is apparently drawn from the Greek word ρεμβω, to turn round and round. He also points out that Archimedes used the term solid rhombus for two right circular cones sharing a common base. For more on the origin of the word, see rhombus at the MathWords web page.