Rhombus

# Rhombus

Overview
In Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, a rhombus or rhomb is a convex quadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...

whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge
Lozenge
A lozenge , often referred to as a diamond, is a form of rhombus. The definition of lozenge is not strictly fixed, and it is sometimes used simply as a synonym for rhombus. Most often, though, lozenge refers to a thin rhombus—a rhombus with acute angles of 45°...

, though the latter sometimes refers specifically to a rhombus with a 45° angle.

Every rhombus is a parallelogram
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...

, and a rhombus with right angles is a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

. (Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

's original definition and some English dictionaries' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition.)

The English word “rhombus” derives from the Ancient Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...

ῥόμβος (rhombos), meaning spinning top.
Discussion

Encyclopedia
In Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, a rhombus or rhomb is a convex quadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...

whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge
Lozenge
A lozenge , often referred to as a diamond, is a form of rhombus. The definition of lozenge is not strictly fixed, and it is sometimes used simply as a synonym for rhombus. Most often, though, lozenge refers to a thin rhombus—a rhombus with acute angles of 45°...

, though the latter sometimes refers specifically to a rhombus with a 45° angle.

Every rhombus is a parallelogram
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...

, and a rhombus with right angles is a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

. (Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

's original definition and some English dictionaries' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition.)

The English word “rhombus” derives from the Ancient Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...

ῥόμβος (rhombos), meaning spinning top. The plural of rhombus can be either rhombi or rhombuses.

## Characterizations

A convex
Convex
'The word convex means curving out or bulging outward, as opposed to concave. Convex or convexity may refer to:Mathematics:* Convex set, a set of points containing all line segments between each pair of its points...

In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...

is a rhombus if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

it is any one of the following:
• a parallelogram with four equal sides
• a parallelogram in which at least two consecutive sides are congruent
• a quadrilateral with four congruent sides
• a parallelogram in which a diagonal bisects an interior angle
• a quadrilateral in which each diagonal bisects two interior angles
• a parallelogram in which the diagonals are perpendicular
• a quadrilateral in which the diagonals are perpendicular and bisect each other

## Properties

Every rhombus has two diagonal
Diagonal
A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...

s connecting pairs of opposite vertices, and two pairs of parallel sides. Using congruent
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...

triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

s, one can prove
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

that the rhombus is symmetric
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

across each of these diagonals. It follows that any rhombus has the following properties:
1. Opposite angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s of a rhombus have equal measure.
2. The two diagonals of a rhombus are perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

; that is, a rhombus is an orthodiagonal quadrilateral
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal to each other....

.
3. Its diagonals bisect opposite angles.

The first property implies that every rhombus is a parallelogram
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...

. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel; adjacent angles are supplementary
Supplementary angles
Supplementary angles are pairs of angles that add up to 180 degrees. Thus the supplement of an angle of x degrees is an angle of degrees....

; the two diagonals bisect
Bisection
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector In geometry, bisection is the division of something into two equal...

one another; any line through the midpoint bisects the area; and the sum of the squares of the sides equals the sum of the squares of the diagonals (the parallelogram law
Parallelogram law
In mathematics, the simplest form of the parallelogram law belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals...

). Thus denoting the common side as a and the diagonals as p and q, every rhombus has

Not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals (the second property) is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite
Kite (geometry)
In Euclidean geometry a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are next to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite each other rather than next to each other...

. Every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus.

A rhombus is a tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral or circumscribed quadrilateral is a convex quadrilateral whose sides all lie tangent to a single circle inscribed within the quadrilateral. This circle is called the incircle...

. That is, it has an inscribed circle that is tangent to all four of its sides.

## Origin

The word rhombus is from the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

word for something that spins. Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

used ρόμβος (rhombos), from the verb ρέμβω (rhembo), meaning "to turn round and round". Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

used the term "solid rhombus" for two right circular cone
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

s sharing a common base.

## In mathematics

• The dual polygon
Dual polygon
In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other.-Properties:Regular polygons are self-dual....

of a rhombus is a rectangle
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

.
• One of the five 2D lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...

types is the rhombic lattice, also called centered rectangular lattice.
• Identical rhombi can tile the 2D plane in three different ways, including, for the 60° rhombus, the Rhombille tiling.
• Three-dimensional analogues of a rhombus include the bipyramid
Bipyramid
An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...

and the bicone
Bicone
A bicone or dicone is the three-dimensional geometric shape swept by revolving an isosceles triangle around its edge of unequal length. Alternatively, one can view a bicone as the surface created by joining two identical right circular cones base-to-base....

.

## Area formulas

As for all parallelograms, the area K of a rhombus is the product of its base and its height. The base is simply any side length a, and the height h is the perpendicular distance between any two non-adjacent sides:

The area can also be expressed as the base squared times the sine of any angle:

or as half the product of the diagonals p, q:

or as the semiperimeter
Semiperimeter
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name...