Төрт бұрышты тіктөртбұрыш

Quadrilateral with four right angles
the record label

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. 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 square. The term "oblong" is used to refer to a non square rectangle. 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 (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 polygons.

De Villiers defines a rectangle more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. 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 perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry 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).

A rectangle is cyclic: all corners lie on a single circle. It is equiangular: all its corner angles are equal (each of 90 degrees). It is isogonal or vertex transitive: all corners lie within the same symmetry orbit. It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).

The dual polygon of a rectangle is a rhombus, as shown in the table below. Rectangle RhombusAll 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. 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 rhombus and vice versa.

A rectangle is a rectilinear polygon: 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 incomparable.

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 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.

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 right triangles. 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 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.