The word rectangle comes from theLatinrectangulus, which is a combination ofrectus (as an adjective, right, proper) andangulus (angle).
Acrossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals[4] (therefore only two sides are parallel). It is a special case of anantiparallelogram, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such asspherical,elliptic, andhyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.
Rectangles are involved in manytiling problems, such as tiling the plane by rectangles or tiling a rectangle bypolygons.
A parallelogram is a special case of a trapezium (known as atrapezoid in North America) in whichboth pairs of opposite sides areparallel andequal inlength.
Star-shaped: 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 withaxes of symmetry through each pair of opposite sides.[9] 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 theperpendicular bisector of those sides, but, in the case of the crossed rectangle, the firstaxis is not an axis ofsymmetry 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 compriseisosceles trapezia and crossed isosceles trapezia (crossed quadrilaterals with the samevertex arrangement as isosceles trapezia).
A rectangle in the plane can be defined by five independentdegrees of freedom consisting, for example, of three for position (comprising two oftranslation and one ofrotation), one for shape (aspect ratio), and one for overall size (area).
Two rectangles, neither of which will fit inside the other, are said to beincomparable.
Formulae
The formula for the perimeter of a rectangleThe area of a rectangle is the product of the length and width.
TheJapanese theorem for cyclic quadrilaterals[12] states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle.
TheBritish flag theorem states that with vertices denotedA,B,C, andD, for any pointP on the same plane of a rectangle:[13]
For every convex bodyC in the plane, we caninscribe a rectangler inC such that ahomothetic copyR ofr is circumscribed aboutC and the positive homothety ratio is at most 2 and.[14]
There exists a unique rectangle with sides and, where is less than, with two ways of being folded along a line through its center such that the area of overlap is minimized and each area yields a different shape – a triangle and a pentagon. The unique ratio of side lengths is.[15]
Crossed rectangles
Acrossedquadrilateral (self-intersecting) consists of two opposite sides of a non-self-intersecting quadrilateral along with the two diagonals. Similarly, a crossed rectangle is acrossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It has the samevertex arrangement as the rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex.
Acrossed quadrilateral is sometimes likened to abow tie orbutterfly, sometimes called an "angular eight". Athree-dimensional rectangularwireframe that is twisted can take the shape of a bow tie.
The interior of acrossed rectangle can have apolygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.
Acrossed rectangle may be consideredequiangular if right and left turns are allowed. As with anycrossed quadrilateral, the sum of itsinterior angles is 720°, allowing for internal angles to appear on the outside and exceed 180°.[16]
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
Asaddle rectangle has 4 nonplanar vertices,alternated from vertices of arectangular cuboid, with a uniqueminimal surface interior defined as a linear combination of the four vertices, creating a saddle surface. This example shows 4 blue edges of the rectangle, and twogreen diagonals, all being diagonal of the cuboid rectangular faces.
Inspherical geometry, aspherical rectangle is a figure whose four edges aregreat 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.
Inelliptic geometry, anelliptic 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.
Inhyperbolic geometry, ahyperbolic 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 periodictessellation patterns, inbrickwork, for example, these tilings:
A perfect rectangle of order 9Lowest-order perfect squared square (1) and the three smallest perfect squared squares (2–4) –all are simple squared squares
A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangulated" (or "triangled") rectangle respectively. The tiled rectangle isperfect[17][18] if the tiles aresimilar and finite in number and no two tiles are the same size. If two such tiles are the same size, the tiling isimperfect. In a perfect (or imperfect) triangled rectangle the triangles must beright triangles. A database of all known perfect rectangles, perfect squares and related shapes can be found atsquaring.net. The lowest number of squares need for a perfect tiling of a rectangle is 9[19] and the lowest number needed for aperfect tilling a square is 21, found in 1978 by computer search.[20]
A rectangle hascommensurable sides if and only if it is tileable by a finite number of unequal squares.[17][21] The same is true if the tiles are unequal isoscelesright triangles.
The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangularpolyominoes, allowing all rotations and reflections. There are also tilings by congruentpolyaboloes.
Unicode
The followingUnicode code points depict rectangles:
U+25AC ▬ BLACK RECTANGLE U+25AD ▭ WHITE RECTANGLE U+25AE ▮ BLACK VERTICAL RECTANGLE U+25AF ▯ WHITE VERTICAL RECTANGLE
^Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, pp. 34–36ISBN1-59311-695-0.
