In
Euclidean plane geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axio ...
, a rectangle is a
quadrilateral with four
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...
s. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a
parallelogram containing a right angle. A rectangle with four sides of equal length is a ''
square''. The term "
oblong" is occasionally used to refer to a non-
square rectangle. A rectangle with
vertices ''ABCD'' would be denoted as .
The word rectangle comes from the
Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
''rectangulus'', which is a combination of ''rectus'' (as an adjective, right, proper) and ''angulus'' (
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
).
A
crossed rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containin ...
is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an
antiparallelogram
In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel. Instead, sides in the ...
, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as
spherical,
elliptic, and
hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.
Rectangles are involved in many
tiling problems, such as tiling the plane by rectangles or tiling a rectangle by
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
s.
Characterizations
A
convex quadrilateral is a rectangle
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
it is any one of the following:
* a
parallelogram with at least one
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...
* a parallelogram with
diagonals of equal length
* a parallelogram ''ABCD'' where
triangles ''ABD'' and ''DCA'' are
congruent
* an equiangular quadrilateral
* a quadrilateral with four right angles
* a quadrilateral where the two diagonals are equal in length and
bisect each other
* a convex quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' whose area is
.
[
* a convex quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' whose area is ]
Classification
Traditional hierarchy
A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
.
A parallelogram is a special case of a trapezium (known as a trapezoid in North America) in which ''both'' pairs of opposite sides are parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster o ...
and equal in length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
.
A trapezium is a convex quadrilateral which has at least one pair of parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster o ...
opposite sides.
A convex quadrilateral is
* Simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
: The boundary does not cross itself.
* Star-shaped
In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This defin ...
: The whole interior is visible from a single point, without crossing any edge.
Alternative hierarchy
De Villiers defines a rectangle more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles. Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, and another which is the perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
for either side that it bisects.
Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia (crossed quadrilaterals with the same vertex arrangement as isosceles trapezia).
Properties
Symmetry
A rectangle is cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
: all corners lie on a single circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
.
It is equiangular: all its corner angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
s are equal (each of 90 degrees).
It is isogonal or vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of fa ...
: all corners lie within the same symmetry orbit
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
.
It has two lines of reflectional symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D th ...
and rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which ...
of order 2 (through 180°).
Rectangle-rhombus duality
The 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.
The dual of an isogonal (vertex-transitive) polygon is an isotoxal (ed ...
of a rectangle is a rhombus, as shown in the table below.
* The figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa.
Miscellaneous
A rectangle is a rectilinear polygon
A rectilinear polygon is a polygon all of whose sides meet at right angles. Thus the interior angle at each vertex is either 90° or 270°. Rectilinear polygons are a special case of isothetic polygons.
In many cases another definition is ...
: its sides meet at right angles.
A rectangle in the plane can be defined by five independent degrees of freedom consisting, for example, of three for position (comprising two of translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
and one of rotation), one for shape ( aspect ratio), and one for overall size (area).
Two rectangles, neither of which will fit inside the other, are said to be incomparable.
Formulae
If a rectangle has length and width
* it has area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
,
* it has perimeter ,
* each diagonal has length ,
* and when , the rectangle is a square.
Theorems
The isoperimetric theorem for rectangles states that among all rectangles of a given perimeter, the square has the largest area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
.
The midpoints of the sides of any quadrilateral with perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
diagonals
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
form a rectangle.
A parallelogram with equal diagonals
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
is a rectangle.
The Japanese theorem for cyclic quadrilaterals
In geometry, the Japanese theorem states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle.
Triangulating an arbitrary cyclic quadrilateral by its diagonals yields four overlapping tr ...
states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle.
The British flag theorem states that with vertices denoted ''A'', ''B'', ''C'', and ''D'', for any point ''P'' on the same plane of a rectangle:
:
For every convex body ''C'' in the plane, we can inscribe a rectangle ''r'' in ''C'' such that a homothetic copy ''R'' of ''r'' is circumscribed about ''C'' and the positive homothety ratio is at most 2 and .
Crossed rectangles
A ''crossed'' ''quadrilateral'' (self-intersecting) consists of two opposite sides of a non-self-intersecting quadrilateral along with the two diagonals. Similarly, a crossed rectangle is a ''crossed quadrilateral'' which consists of two opposite sides of a rectangle along with the two diagonals. It has the same vertex arrangement as the rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex.
A ''crossed quadrilateral'' is sometimes likened to a bow tie or butterfly
Butterflies are insects in the macrolepidopteran clade Rhopalocera from the order Lepidoptera, which also includes moths. Adult butterflies have large, often brightly coloured wings, and conspicuous, fluttering flight. The group compris ...
, sometimes called an "angular eight". A three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
rectangular wire
Overhead power cabling. The conductor consists of seven strands of steel (centre, high tensile strength), surrounded by four outer layers of aluminium (high conductivity). Sample diameter 40 mm
A wire is a flexible strand of metal.
Wire is co ...
frame that is twisted can take the shape of a bow tie.
The interior of a ''crossed rectangle'' can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.
A ''crossed rectangle'' may be considered equiangular if right and left turns are allowed. As with any ''crossed quadrilateral'', the sum of its interior angles is 720°, allowing for internal angles to appear on the outside and exceed 180°.
A rectangle and a crossed rectangle are quadrilaterals with the following properties in common:
* Opposite sides are equal in length.
* The two diagonals are equal in length.
* It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).
Other rectangles
In spherical geometry, a spherical rectangle is a figure whose four edges are great circle arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. The surface of a sphere in Euclidean solid geometry is a non-Euclidean surface in the sense of elliptic geometry. Spherical geometry is the simplest form of elliptic geometry.
In elliptic geometry, an elliptic rectangle is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length.
In hyperbolic geometry, a hyperbolic rectangle is a figure in the hyperbolic plane whose four edges are hyperbolic arcs which meet at equal angles less than 90°. Opposite arcs are equal in length.
Tessellations
The rectangle is used in many periodic tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
patterns, in brickwork, for example, these tilings:
Squared, perfect, and other tiled rectangles
A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangulated" (or "triangled") rectangle respectively. The tiled rectangle is ''perfect'' if the tiles are similar and finite in number and no two tiles are the same size. If two such tiles are the same size, the tiling is ''imperfect''. In a perfect (or imperfect) triangled rectangle the triangles must be right triangles. A database of all known perfect rectangles, perfect squares and related shapes can be found a
squaring.net
The lowest number of squares need for a perfect tiling of a rectangle is 9 and the lowest number needed for a perfect tilling a square is 21, found in 1978 by computer search.
A rectangle has commensurable sides if and only if it is tileable by a finite number of unequal squares. The same is true if the tiles are unequal isosceles right triangles.
The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular polyominoes, allowing all rotations and reflections. There are also tilings by congruent polyaboloes.
Unicode
U+25AC ▬ BLACK RECTANGLE
U+25AD ▭ WHITE RECTANGLE
U+25AE ▮ BLACK VERTICAL RECTANGLE
U+25AF ▯ WHITE VERTICAL RECTANGLE
See also
* Cuboid
* Golden rectangle
* Hyperrectangle
In geometry, an orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions.
A necessary and sufficient condition is that it is congruent to the Cartesian product of intervals. If al ...
* Superellipse (includes a rectangle with rounded corners)
References
External links
*
Definition and properties of a rectangle
with interactive animation.
with interactive animation.
{{Authority control
Types of quadrilaterals
Elementary shapes