# Rectangle

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Rectangle
Rectangle
Edges and vertices 4
Schläfli symbol { } × { }
Coxeter diagram
Symmetry group Dihedral (D2), [2], (*22), order 4
Dual polygon rhombus
Properties convex, isogonal, cyclic Opposite angles and sides are congruent

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°). 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 oblong is occasionally used to refer to a non-square rectangle. [1] [2] [3] A rectangle with vertices ABCD would be denoted as  ABCD.

## Contents

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. [4] It is a special case of an antiparallelogram, and its angles are not right angles. 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.

## Characterizations

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
• 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]

## Classification

A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular.

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 quadrilateral which has at least one pair of parallel opposite sides.

• Simple : The boundary does not cross itself.
• Star-shaped : The whole interior is visible from a single point, without crossing any edge.

### Alternative hierarchy

De Villiers defines a rectangle more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. [9] 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).

## Properties

### Symmetry

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°).

### Rectangle-rhombus duality

The dual polygon of a rectangle is a rhombus, as shown in the table below. [10]

RectangleRhombus
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.
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.

### Miscellaneous

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 incomparable.

## Formulae

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.

## Theorems

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]

## Crossed rectangles

A crossed (self-intersecting) quadrilateral 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. A three-dimensional rectangular wire 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 interior angles (two acute and two reflex), as with any crossed quadrilateral, is 720°. [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°).

## Other rectangles

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.

## Tessellations

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

## Squared, perfect, and other tiled rectangles

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 rectangle has commensurable sides if and only if it is tileable by a finite number of unequal squares. [15] [17] 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.

## Related Research Articles

A quadrilateral is a polygon in Euclidean plane geometry with four edges (sides) and four vertices (corners). Other names for quadrilateral include quadrangle, tetragon, and 4-gon. A quadrilateral with vertices , , and is sometimes denoted as .

In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other instead of being adjacent. Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape and which are in turn named for a bird. Kites are also known as deltoids, but the word "deltoid" may also refer to a deltoid curve, an unrelated geometric object.

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector.

In plane Euclidean geometry, a rhombus is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezium in English outside North America, but as a trapezoid in American and Canadian English. The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides. A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast to the special cases below.

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, who is said to have offered an ox, probably to the god Apollo, as a sacrifice of thanksgiving for the discovery, but it is sometimes attributed to Pythagoras.

In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of the same measure. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides are parallel, and the two other sides are of equal length. The diagonals are also of equal length. The base angles of an isosceles trapezoid are equal in measure.

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ABCD.

In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths.

Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, whose proof was published posthumously in 1731.

In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. It has also been called an exscriptible quadrilateral. The circle is called its excircle, its radius the exradius and its center the excenter. The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The ex-tangential quadrilateral is closely related to the tangential quadrilateral.

In Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified first according to whether they were equidiagonal and then into more specialized types.

In Euclidean geometry, a right kite is a kite that can be inscribed in a circle. That is, it is a kite with a circumcircle. Thus the right kite is a convex quadrilateral and has two opposite right angles. If there are exactly two right angles, each must be between sides of different lengths. All right kites are bicentric quadrilaterals, since all kites have an incircle. One of the diagonals divides the right kite into two right triangles and is also a diameter of the circumcircle.

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11. Cyclic Quadrilateral Incentre-Rectangle 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.
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