Rectangle

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Rectangle

Type	Quadrilateral
Edges and vertices	4
Schläfli symbol	{}x{}
Symmetry group	D₂ (*2)
Coxeter-Dynkin diagram
Dual polygon	Rhombus
Properties	convex, isogonal, cyclic

In Euclidean geometry, the term rectangle normally refers to a quadrilateral with four right angles. This is a simple rectangle.

A rectangle that is not simple is complex, but more clearly described as self-intersecting or crossed. It is defined as a self-intersecting quadrilateral with the same vertex arrangement as a simple rectangle.

In recreational mathematics a popular subject is the tiling of rectangles by polygons, ranging from simple puzzles to unsolved problems.

1 Properties
2 Related polygons
3 Area, perimeter, and other facts
4 Tessellations
5 Crossed rectangle
6 Squared, perfect, and other tiled rectangles
7 See also
8 References
9 External links

[ Properties

All angles are 90 degrees.
Opposite sides are equal in length.
Opposite sides are parallel.
Diagonals are equal in length and bisect each other.

[ Related polygons

A rectangle is a cyclic polygon.
The dual polygon of a rectangle is a rhombus.
When the length is equal to the width, the rectangle is a square.
A rectangle is a special case of a parallelogram, which has two pairs of parallel opposite sides. A parallelogram, and hence also a rectangle, is a special case of a trapezium (known as a trapezoid in North America), which has at least one pair of parallel opposite sides.

[ Area, perimeter, and other facts

The formula for the perimeter of a rectangle.

If a rectangle has length $l$ and width $w$

it has area $A = l w$
perimeter $P = 2 l + 2 w = 2(l + w)$
and each diagonal has length $\sqrt{l^2 + w^2}$ .

The term oblong is occasionally used to refer to a non-square rectangle. ^[1]^[2]

Two rectangles, neither of which will fit inside the other, are said to be incomparable.

[ Tessellations

The rectangle is used in many periodic tessellation patterns, in brickwork, for example, these isogonal tilings:

Stacked bond	Running bond	Basket weave	Basket weave	Herringbone pattern

[ Crossed rectangle

A crossed rectangle is a complex (self-intersecting) rectangle, also called a bow-tie rectangle or butterfly rectangle.

It has the same vertex arrangement as a simple rectangle with which it shares two edges. Its other two edges are the diagonals of the simple rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex.

The interior of a crossed rectangle can have a polygon density of +/-1 in each half triangle, dependent upon the winding orientation as clockwise or counterclockwise.

[ Squared, perfect, and other tiled rectangles

A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangled" (or "triangulated") rectangle respectively. The tiled rectangle is perfect[3]^[4] 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 tilable by a finite number of unequal squares.^[5]^[3] 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.