Rhombus
From Wikipedia, the free encyclopedia
- For other uses of the word rhombus, see Rhombus (disambiguation)
In geometry, a rhombus (or rhomb) is a quadrilateral in which all of the sides are of equal length, i.e., it is an equilateral quadrangle. In colloquial usage the shape is often described as a diamond or lozenge.
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[edit] Two definitions
If a figure meeting the above description has a right angle, then all its angles are right angles and it is a square. There is disagreement as to whether a square is considered a kind of rhombus or not. If it is, the above definition of rhombus is the complete one; if not, the definition additionally requires the angles not to be right angles.
[edit] Properties
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.
The rhombus has the different symmetry as the rectangle (with symmetry group D2, the Klein four-group) 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 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.
[edit] Proof
[edit] The diagonals are perpendicular.
Let A, B, C and D be the vertices of the rhombus, named in agreement with the figure (higher on this page). 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,
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:
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):
[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.
[edit] External links
- Rhombus definition. Math Open Reference With interactive applet.
- Rhombus area. Math Open Reference Shows three different ways to compute the area of a rhombus, with interactive applet.
