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rectangle
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Characterizations
A convex quadrilateral is a rectangle if and only if it is any one of the following:Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, pp. 34â€“36 {{isbn|1-59311-695-0}}.BOOK, Owen Byer, Felix Lazebnik, Deirdre L. Smeltzer, Methods for Euclidean Geometry,weblink 2011-11-13, 19 August 2010, MAA, 978-0-88385-763-2, 53â€“,- a parallelogram with at least one right angle
- 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 convex quadrilateral with successive sides a, b, c, d whose area is tfrac{1}{4}(a+c)(b+d).{{rp|fn.1}}
- a convex quadrilateral with successive sides a, b, c, d whose area is tfrac{1}{2} sqrt{(a^2+c^2)(b^2+d^2)}.JOURNAL, Josefsson Martin, 2013, Five Proofs of an Area Characterization of Rectangles,weblink PDF, Forum Geometricorum, 13, 17â€“21,
Classification
File:Symmetries_of_square.svg|280px|thumb|A rectangle is a special case of both parallelogram and trapezoid. A squaresquareTraditional hierarchy
A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular.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 and equal in length.A trapezium is a convex quadrilateral which has at least one pair of parallel opposite sides.A convex quadrilateral is- Simple: The boundary does not cross itself.
- 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 with axes of symmetry through each pair of opposite sides.An Extended Classification of Quadrilaterals (An excerpt from De Villiers, M. 1996. Some Adventures in Euclidean Geometry. University of Durban-Westville.) 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 bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry 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: all corners lie on a single circle.It is equiangular: all its corner angles are equal (each of 90 degrees).It is isogonal or vertex-transitive: all corners lie within the same symmetry orbit.It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180Â°).Rectangle-rhombus duality
The dual polygon of a rectangle is a rhombus, as shown in the table below.de Villiers, Michael, "Generalizing Van Aubel Using Duality", Mathematics Magazine 73 (4), Oct. 2000, pp. 303-307.{|class="wikitable" style="text-align:center"!Rectangle !! Rhombus|All angles are equal.All sides are equal.|Alternate sides are equal.Alternate angles are equal.Vertex (geometry)>vertices, hence it has a circumcircle.Its centre is equidistant from its sides, hence it has an incircle.|Two axes of symmetry bisect opposite sides.Two axes of symmetry bisect opposite angles.|Diagonals are equal in length.Diagonals intersect at equal angles.- The figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa.
Miscellaneous
The two diagonals are equal in length and bisect each other. Every quadrilateral with both these properties is a rectangle.A rectangle is rectilinear: 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 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
(File:PerimeterRectangle.svg|thumb|150px|The formula for the perimeter of a rectangle)(File:Illustration for the area of a rectangle.svg|thumb|150px|The area of a rectangle is the product of the length and width.)If a rectangle has length ell and width w- it has area A = ell w,,
- it has perimeter P = 2ell + 2w = 2(ell + w),,
- each diagonal has length d=sqrt{ell^2 + w^2},
- and when ell = w,, 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.The midpoints of the sides of any quadrilateral with perpendicular diagonals form a rectangle.A parallelogram with equal diagonals is a rectangle.The Japanese theorem for cyclic quadrilateralsCyclic Quadrilateral Incentre-Rectangle with interactive animation illustrating a rectangle that becomes a 'crossed rectangle', making a good case for regarding a 'crossed rectangle' as a type of rectangle. 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:JOURNAL, Hall, Leon M., Robert P. Roe, yes, An Unexpected Maximum in a Family of Rectangles, Mathematics Magazine, 71, 4, 1998, 285â€“291,weblink 2690700,
displaystyle (AP)^2 + (CP)^2 = (BP)^2 + (DP)^2.
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 0.5 text{ Ã— Area}(R) leq text{Area}(C) leq 2 text{ Ã— Area}(r).JOURNAL, 10.1007/BF01263495, Approximation of convex bodies by rectangles, Geometriae Dedicata, 47, 111, 1993, Lassak, M., Crossed rectangles
A crossed (self-intersecting) quadrilateral 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. A three-dimensional rectangular wire frame that is twisted can take the shape of a bow tie. A crossed rectangle is sometimes called an "angular eight".The interior of a crossed rectangle can have a polygon density of Â±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.A crossed rectangle is not equiangular. The sum of its interior angles (two acute and two reflex), as with any crossed quadrilateral, is 720Â°.Stars: A Second Look. (PDF). Retrieved 2011-11-13.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 180Â°).
Other rectangles
File:Saddle rectangle example.png|thumb|A saddle rectangle has 4 nonplanar vertices, alternated from vertices of a cuboid, with a unique minimal 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 two greengreenIn spherical geometry, a spherical rectangle is a figure whose four edges are great circle arcs which meet at equal angles greater than 90Â°. 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 90Â°. 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 90Â°. Opposite arcs are equal in length.Tessellations
The rectangle is used in many periodic tessellation patterns, in brickwork, for example, these tilings:{|class=wikitable align=center182px)Stacked bond150px)Running bond150px)Basket weave150px)Basket weave150px)Herringbone patternSquared, 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 perfectJOURNAL, J.D. Skinner II, C.A.B. Smith and W.T. Tutte, November 2000, On the Dissection of Rectangles into Right-Angled Isosceles Triangles, Journal of Combinatorial Theory, Series B, 80, 2, 277â€“319, 10.1006/jctb.2000.1987, 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 rectangle has commensurable sides if and only if it is tileable by a finite number of unequal squares.JOURNAL, R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte, 1940, The dissection of rectangles into squares, Duke Mathematical Journal, Duke Math. J., 7, 1, 312â€“340, 10.1215/S0012-7094-40-00718-9,weblink JOURNAL, R. Sprague, Å¨ber die Zerlegung von Rechtecken in lauter verschiedene Quadrate, Crelle's Journal, Journal fÃ¼r die reine und angewandte Mathematik, 182, 1940, 60â€“64, The same is true if the tiles are unequal isosceles (wikt:right triangle|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.See also
- Cuboid
- Golden rectangle
- Hyperrectangle
- Superellipse (includes a rectangle with rounded corners)
References
{{reflist|35em}}External links
{{Commons category|Rectangles}}- {{MathWorld |urlname=Rectangle |title=Rectangle}}
- Definition and properties of a rectangle with interactive animation.
- Area of a rectangle with interactive animation.
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