Rhombus

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For other uses, see Rhombus (disambiguation).

Two rhombi.

In geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos, “rhombus, spinning top”), (plural rhombi or rhombuses) or rhomb (plural rhombs) is an equilateral quadrilateral. In other words, it is a four-sided polygon in which every side has the same length.

The rhombus is often casually called a diamond, after the diamonds suit in playing cards, or a lozenge, because those shapes are rhombi (though not all rhombi are actually diamonds or lozenges).

1 Supersets
2 Area
3 A proof that the diagonals are perpendicular
4 Origin
5 References
6 External links

Supersets

In any rhombus, opposite sides are parallel. Thus, the rhombus is a special case of the parallelogram. One analogy holds that the rhombus is to the parallelogram as the square is to the rectangle.

A rhombus is also a special case of a kite (a quadrilateral with two distinct pairs of adjacent sides of equal lengths). The opposite sides of a kite are not parallel unless the kite is also a rhombus.

Area

The area of any rhombus is half the product of the lengths of its diagonals:

Failed to parse (Missing texvc executable; please see math/README to configure.): Area=\frac{D_1 \times D_2}{2}

Because the rhombus is a parallelogram, the area also equals the length of a side (B) multiplied by the perpendicular distance between two opposite sides(H)

Failed to parse (Missing texvc executable; please see math/README to configure.): Area=B \times H

The area also equals the square of the side multiplied by the sine of any of the interior angles:

Failed to parse (Missing texvc executable; please see math/README to configure.): Area={a^2 \sin\theta}

where a is the length of the side and Failed to parse (Missing texvc executable; please see math/README to configure.): \theta

is the angle between two sides.

A proof that the diagonals are perpendicular

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

If A, B, C and D were the vertices of the rhombus, named in agreement with the figure (higher on this page). Using Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{AB}

to represent the vector from A to B, one notices that

Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}

Failed to parse (Missing texvc executable; please see math/README to configure.): \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,

Failed to parse (Missing texvc executable; please see math/README to configure.): <\overrightarrow{AC}, \overrightarrow{BD}> = <\overrightarrow{AB} + \overrightarrow{BC}, \overrightarrow{BC} - \overrightarrow{AB}>

Failed to parse (Missing texvc executable; please see math/README to configure.): = <\overrightarrow{AB}, \overrightarrow{BC}> - <\overrightarrow{AB}, \overrightarrow{AB}> + <\overrightarrow{BC}, \overrightarrow{BC}> - <\overrightarrow{BC}, \overrightarrow{AB}>

Failed to parse (Missing texvc executable; please see math/README to configure.): = 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.

Origin

The word rhombus is from the Greek word for something that spins. Euclid used ρόμβος (rhombos), from the verb ρέμβω (rhembo), meaning "to turn round and round" ^[1]^[2]. Archimedes used the term "solid rhombus" for two right circular cones sharing a common base.^[3]