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.

Contents 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

[edit] 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.

[edit] 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.

[edit] Area, perimeter, and other facts

If a rectangle has length l and width w

• it has area A = lw
• perimeter P = 2l + 2w = 2(l + w)
• and each diagonal has length .

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.

[edit] 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

[edit] 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.

[edit] 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.

[edit] See also

• Square
• Cuboid
• Hyperrectangle
• Golden rectangle

[edit] References

[edit] 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