Rhombus
In
geometry, a rhombus 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.
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. If any angle of a rhombus is a right angle, then all the angles of that rhombus are right angles, and it is then a rectangle and a square.
Encyclopedia
In
geometry, a
rhombus 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.
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. If any angle of a rhombus is a right angle, then all the angles of that rhombus are right angles, and it is then a rectangle and a square. 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 same
symmetry as the rectangle 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.
Consecutive angles of a rhombus are supplementary.
Proof
The diagonals are perpendicular.
Let A, B, C and D be the vertices of the rhombus, named in agreement with the figure . 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.
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 multiplied by the perpendicular distance between two opposite sides:
Origin
The origin of the word
rhombus is from the Greek word for something that spins.
Euclid uses the word ??µß??; and in his translation Heath says it is apparently drawn from the Greek word ?eµß?, 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 .
External links
- With interactive applet.
- Shows three different ways to compute the area of a rhombus, with interactive applet.