rectangle

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

Rectangles may be used in periodic tilings of the plane. Another popular subject in recreational mathematics is the tiling of rectangles by polygons, ranging from simple puzzles to unsolved problems.

General properties

Every rectangle, simple or crossed, has the following properties:

• It is isogonal.
• The two diagonals are equal in length.
• Opposite sides are equal in length.
• The dual polygon of a rectangle is a rhombus.

Simple rectangle

A simple rectangle has the following properties:

• It is convex and cyclic.
• All angles are 90 degrees.
• The two diagonals bisect each other.
• Opposite sides are parallel.

A simple 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.

The formula for the perimeter of a rectangle.

If a simple rectangle has length l and width w

• it has area A = lw,
• it has perimeter P = 2l + 2w = 2(l + w),
• each diagonal has length ,
• and when l = w, the rectangle is a square.

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

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

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. To make a text rectangle hold down alt and press 2267 while alt is held down and you will get this image : █

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

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.

See also

• Cuboid
• Hyperrectangle
• Golden rectangle

References

External links

Wikimedia Commons has media related to category:      Rectangles

• Weisstein, Eric W., "Rectangle" from MathWorld.
• Definition and properties of a rectangle With interactive animation
• Area of a rectangle with interactive animation