Rhombus

Rhombus

[rom-buhs]

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).

Area

The area of any rhombus is the product of the lengths of its diagonals divided by two:

Area=({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)

Area=B times H

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

Area=a^2 sintheta

where a is the length of the side and 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 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,

=
= - + -
= 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.

Tilings


Rhombic tiling

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". Archimedes used the term "solid rhombus" for two right circular cones sharing a common base.

This is also a called Tessellation.

References

External links

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