In Euclidean plane geometry, a rectangle is a quadrilateral with four . It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term is occasionally used to refer to a nonsquare rectangle.http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf Definition of Oblong. Mathsisfun.com. Retrieved 20111113. Oblong – Geometry – Math Dictionary. Icoachmath.com. Retrieved 20111113. A rectangle with vertices ABCD would be denoted as .
The word rectangle comes from the Latin rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus (angle).
A crossed rectangle is a crossed (selfintersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a special case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have socalled 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 .
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 polygon quadrilateral which has at least one pair of parallel opposite sides.
A convex quadrilateral is
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).
It is equiangular: all its corner are equal (each of 90 degrees).
It is isogonal or vertextransitive: all corners lie within the same symmetry orbit.
It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).
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 circumcircle.  Its centre is equidistant from its sides, hence it has an incircle. 
Its axes of symmetry bisect opposite sides.  Its axes of symmetry bisect opposite angles. 
Diagonals are equal in length.  Diagonals intersect at equal angles. 
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 Comparability.
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 quadrilaterals Cyclic Quadrilateral IncentreRectangle 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:
For every convex body C in the plane, we can Inscribed figure 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 $0.5\; \backslash text\{\; \times \; Area\}(R)\; \backslash leq\; \backslash text\{Area\}(C)\; \backslash leq\; 2\; \backslash text\{\; \times \; Area\}(r)$.
A crossed quadrilateral is sometimes likened to a bow tie or butterfly. A threedimensional rectangular wire Space 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 ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.
A crossed rectangle is not equiangular. The sum of its (two acute and two Reflex angle), as with any crossed quadrilateral, is 720°. Stars: A Second Look. (PDF). Retrieved 20111113.
A rectangle and a crossed rectangle are quadrilaterals with the following properties in common:
In spherical geometry, a spherical rectangle is a figure whose four edges are great circle 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 nonEuclidean 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.
Stacked bond 
Running bond 
Basket weave  Basket weave  Herringbone pattern 
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 nonrectangular , allowing all rotations and reflections. There are also tilings by congruent .
