Rhombus
From Wikipedia, the free encyclopedia
In geometry, a rhombus (plural rhombi) 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).
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[edit] 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 pairs of equal adjacent sides). The opposite sides of a kite are not parallel unless the kite is also a rhombus.
[edit] Area
The area of any rhombus is half the product of the lengths of its diagonals:
<math>A=\frac{D_1 \times D_2}{2}</math>
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):
<math>A=B \times H</math>
The area also equals the square of the side multiplied by the sine of any of the interior angles:
<math>A={a^2 \sin\theta}</math>
where a is the length of the side and <math>\theta</math> is the angle between two sides.
[edit] 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 <math>\overrightarrow{AB}</math> to represent the vector from A to B, one notices that
<math>\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}</math>
<math>\overrightarrow{BD} = \overrightarrow{BC}+ \overrightarrow{CD}= \overrightarrow{BC}- \overrightarrow{AB}</math>.
The last equality comes from the parallelism of CD and AB.
Taking the inner product,
- <math>
<\overrightarrow{AC}, \overrightarrow{BD}> = <\overrightarrow{AB} + \overrightarrow{BC}, \overrightarrow{BC} - \overrightarrow{AB}></math>
- <math>= <\overrightarrow{AB}, \overrightarrow{BC}> - <\overrightarrow{AB}, \overrightarrow{AB}> + <\overrightarrow{BC}, \overrightarrow{BC}> - <\overrightarrow{BC}, \overrightarrow{AB}></math>
- <math> = 0</math>
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] 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
- Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)
- 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.ar:دالتون
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