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# Rectangle  ( Types Of Quadrilaterals )

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In Euclidean plane geometry, a rectangle is a with four . 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 . The term is occasionally used to refer to a non- rectangle. Definition of Oblong. Mathsisfun.com. Retrieved 2011-11-13. Oblong – Geometry – Math Dictionary. Icoachmath.com. Retrieved 2011-11-13. A rectangle with vertices ABCD would be denoted as .

The word rectangle comes from the rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus ().

A crossed rectangle 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, 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 .

Characterizations
A 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. 34–36 .
(2010). 9780883857632, MAA. .
• a with at least one
• a parallelogram with of equal length
• a parallelogram ABCD where ABD and DCA are congruent
• a quadrilateral with four right angles
• a quadrilateral where the two diagonals are equal in length and each otherGerard Venema, "Exploring Advanced Euclidean Geometry with GeoGebra", MAA, 2013, p. 56.
• a convex quadrilateral with successive sides a, b, c, d whose area is $\tfrac\left\{1\right\}\left\{4\right\}\left(a+c\right)\left(b+d\right)$.
• a convex quadrilateral with successive sides a, b, c, d whose area is $\tfrac\left\{1\right\}\left\{2\right\} \sqrt\left\{\left(a^2+c^2\right)\left(b^2+d^2\right)\right\}.$

Classification

A rectangle is a special case of a in which each pair of adjacent sides is .

A parallelogram is a special case of a trapezium (known as a in North America) in which both pairs of opposite sides are parallel and equal in .

A trapezium is a which has at least one pair of parallel opposite sides.

• : 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 bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of 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 : all lie on a single .

It is equiangular: all its corner are equal (each of 90 degrees).

It is isogonal or vertex-transitive: all corners lie within the same .

It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).

Rectangle-rhombus duality
The of a rectangle is a , as shown in the table below.de Villiers, Michael, "Generalizing Van Aubel Using Duality", Mathematics Magazine 73 (4), Oct. 2000, pp. 303-307.

All angles are equal.All sides are equal.
Alternate sides are equal.Alternate angles are equal.
Its centre is equidistant from its vertices, hence it has a .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 and vice versa.

Miscellaneous
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 ), 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 .

Formulae
If a rectangle has length $\ell$ and width $w$
• it has $A = \ell w\,$,
• it has $P = 2\ell + 2w = 2\left(\ell + w\right)\,$,
• each diagonal has length $d=\sqrt\left\{\ell^2 + w^2\right\}$,
• and when $\ell = w\,$, the rectangle is a square.

Theorems
The isoperimetric theorem for rectangles states that among all rectangles of a given , the square has the largest .

The midpoints of the sides of any with form a rectangle.

A with equal is a rectangle.

The Japanese theorem for cyclic quadrilaterals Cyclic 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:

$\displaystyle \left(AP\right)^2 + \left(CP\right)^2 = \left(BP\right)^2 + \left(DP\right)^2.$

For every convex body C in the plane, we can 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 $0.5 \text\left\{ × Area\right\}\left(R\right) \leq \text\left\{Area\right\}\left(C\right) \leq 2 \text\left\{ × Area\right\}\left(r\right)$.

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 or , sometimes called an "angular eight". A three-dimensional rectangular that is twisted can take the shape of a bow tie.

The interior of a crossed rectangle can have a 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 is 720°, allowing for internal angles to appear on the outside and exceed 180°. 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 180°).

Other rectangles
In spherical geometry, a spherical rectangle is a figure whose four edges are 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 patterns, in , for example, these tilings:

Stacked bond

Running bond

Herringbone pattern

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 . A database of all known perfect rectangles, perfect squares and related shapes can be found at 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 .

The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular , allowing all rotations and reflections. There are also tilings by congruent .

Unicode
   U+25AC ▬ BLACK RECTANGLE
U+25AE ▮ BLACK VERTICAL RECTANGLE
U+25AF ▯ WHITE VERTICAL RECTANGLE


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