Rectangle

A convex quadrilateral is a rectangle if and only if it is any one of the following:[5][6]

• a parallelogram with at least one right angle
• a parallelogram with diagonals of equal length
• a parallelogram ABCD where triangles ABD and DCA are congruent
• an equiangular quadrilateral
• a quadrilateral with four right angles
• a quadrilateral where the two diagonals are equal in length and bisect each other[7]
• a convex quadrilateral with successive sides a, b, c, d whose area is ${\displaystyle {\tfrac {1}{4}}(a+c)(b+d)}$.[8]^: fn.1
• a convex quadrilateral with successive sides a, b, c, d whose area is ${\displaystyle {\tfrac {1}{2}}{\sqrt {(a^{2}+c^{2})(b^{2}+d^{2})}}.}$[8]

The formula for the perimeter of a rectangle

The area of a rectangle is the product of the length and width.

If a rectangle has length ${\displaystyle \ell }$ and width ${\displaystyle w}$

• it has area ${\displaystyle A=\ell w\,}$,
• it has perimeter ${\displaystyle P=2\ell +2w=2(\ell +w)\,}$,
• each diagonal has length ${\displaystyle d={\sqrt {\ell ^{2}+w^{2}}}}$,
• and when ${\displaystyle \ell =w\,}$, the rectangle is a square.

The isoperimetric theorem for rectangles states that among all rectangles of a given perimeter, the square has the largest area.

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[11] 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:[12]

${\displaystyle \displaystyle (AP)^{2}+(CP)^{2}=(BP)^{2}+(DP)^{2}.}$

For every convex body C in the plane, we can inscribe 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 ${\displaystyle 0.5{\text{ × Area}}(R)\leq {\text{Area}}(C)\leq 2{\text{ × Area}}(r)}$.[13]

A crossed quadrilateral (self-intersecting) consists of two opposite sides of a non-self-intersecting quadrilateral along with the two diagonals. Similarly, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It has the same vertex arrangement as the rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex.

A crossed quadrilateral is sometimes likened to a bow tie or butterfly, sometimes called an "angular eight". A three-dimensional rectangular wire frame that is twisted can take the shape of a bow tie.

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 may be considered equiangular if right and left turns are allowed. As with any crossed quadrilateral, the sum of its interior angles is 720°, allowing for internal angles to appear on the outside and exceed 180°.[14]

A rectangle and a crossed rectangle are quadrilaterals with the following properties in common:

• Opposite sides are equal in length.
• The two diagonals are equal in length.
• It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).

A saddle rectangle has 4 nonplanar vertices, alternated from vertices of a cuboid, with a unique minimal surface interior defined as a linear combination of the four vertices, creating a saddle surface. This example shows 4 blue edges of the rectangle, and two green diagonals, all being diagonal of the cuboid rectangular faces.

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.

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

Stacked bond

Running bond

Basket weave

Basket weave

Herringbone pattern

A perfect rectangle of order 9

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[15][16] 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[17] and the lowest number needed for a perfect tilling a square is 21, found in 1978 by computer search.[18]

A rectangle has commensurable sides if and only if it is tileable by a finite number of unequal squares.[15][19] 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.

   U+25AC ▬ BLACK RECTANGLE
   U+25AD ▭ WHITE RECTANGLE
   U+25AE ▮ BLACK VERTICAL RECTANGLE
   U+25AF ▯ WHITE VERTICAL RECTANGLE

• Cuboid
• Golden rectangle
• Hyperrectangle
• Superellipse (includes a rectangle with rounded corners)

Wikimedia Commons has media related to Rectangles.

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