In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a square is a
regular quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
. It has four straight sides of equal length and four equal
angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
s. Squares are special cases of
rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
s, which have four equal angles, and of
rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) 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 rhom ...
es, which have four equal sides. As with all rectangles, a square's angles are
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
s (90
degrees, or
/2
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s), making adjacent sides
perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
. The
area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of a square is the side length multiplied by itself, and so in
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, multiplying a number by itself is called
squaring.
Equal squares can tile the plane edge-to-edge in the
square tiling. Square tilings are ubiquitous in
tile
Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, Rock (geology), stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, wal ...
d floors and walls,
graph paper, image
pixel
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a Raster graphics, raster image, or the smallest addressable element in a dot matrix display device. In most digital display devices, p ...
s, and
game boards. Square shapes are also often seen in building
floor plans,
origami paper, food servings, in
graphic design
Graphic design is a profession, academic discipline and applied art that involves creating visual communications intended to transmit specific messages to social groups, with specific objectives. Graphic design is an interdisciplinary branch of ...
and
heraldry
Heraldry is a discipline relating to the design, display and study of armorial bearings (known as armory), as well as related disciplines, such as vexillology, together with the study of ceremony, Imperial, royal and noble ranks, rank and genealo ...
, and in instant photos and fine art.
The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for
squaring the circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square (geometry), square with the area of a circle, area of a given circle by using only a finite number of steps with a ...
by
compass and straightedge
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...
, now known to be impossible. Squares can be inscribed in any smooth or convex curve such as a circle or triangle, but it remains unsolved
whether a square can be inscribed in every simple closed curve. Several problems of
squaring the square involve subdividing squares into unequal squares. Mathematicians have also studied packing squares as tightly as possible into other shapes.
Squares can be constructed by
straightedge and compass, through their
Cartesian coordinates, or by repeated multiplication by
in the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
. They form the
metric balls for
taxicab geometry and
Chebyshev distance, two forms of non-Euclidean geometry. Although
spherical geometry and
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
both lack polygons with four equal sides and right angles, they have square-like regular polygons with four sides and other angles, or with right angles and different numbers of sides.
Definitions and characterizations

Squares can be defined or characterized in many equivalent ways. If a
polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
in the
Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
satisfies any one of the following criteria, it satisfies all of them:
* A square is a polygon with four equal sides and four
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
s; that is, it is a quadrilateral that is both a rhombus and a rectangle
[
* A square is a ]rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
with four equal sides.
* A square is a rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) 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 rhom ...
with a right angle between a pair of adjacent sides.[
* A square is a rhombus with all angles equal.][
* A square is a ]parallelogram
In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...
with one right angle and two adjacent equal sides.[
* A square is a quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other. That is, it is a rhombus with equal diagonals.
* A square is a quadrilateral with successive sides , , , whose area is
Squares are the only ]regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
s whose internal angle, central angle, and external angle are all equal (they are all right angles).[
]
Properties
A square is a special case of a rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) 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 rhom ...
(equal sides, opposite equal angles), a kite
A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...
(two pairs of adjacent equal sides), a trapezoid (one pair of opposite sides parallel), a parallelogram
In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...
(all opposite sides parallel), a quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
or tetragon (four-sided polygon), and a rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
(opposite sides equal, right-angles),[ and therefore has all the properties of all these shapes, namely:
* All four internal angles of a square are equal (each being 90°, a right angle).]
* The central angle of a square is equal to 90°.
* The external angle of a square is equal to 90°.[
* The diagonals of a square are equal and bisect each other, meeting at 90°.][
* The diagonals of a square bisect its internal angles, forming adjacent angles of 45°.
* All four sides of a square are equal.
* Opposite sides of a square are parallel.
All squares are similar to each other, meaning they have the same shape. One parameter (typically the length of a side or diagonal) suffices to specify a square's size. Squares of the same size are congruent.
]
Measurement
A square whose four sides have length has perimeter
A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimet ...
and diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek � ...
length . The square root of 2
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...
, appearing in this formula, is irrational, meaning that it is not the ratio of any two integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s. It is approximately equal to 1.414, and its approximate value was already known in Babylonian mathematics. A square's area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
is
This formula for the area of a square as the second power of its side length led to the use of the term '' squaring'' to mean raising any number to the second power. Reversing this relation, the side length of a square of a given area is the square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
of the area. Squaring an integer, or taking the area of a square with integer sides, results in a square number; these are figurate numbers representing the numbers of points that can be arranged into a square grid.
Since four squared equals sixteen, a four by four square has an area equal to its perimeter. That is, it is an equable shape. The only other equable integer rectangle is a three by six rectangle.
Because it is a regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if ''A'' and ''P'' are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds:
with equality if and only if the quadrilateral is a square.
Symmetry
The square is the most symmetrical of the quadrilaterals.[ Eight rigid transformations of the plane take the square to itself:]
For an axis-parallel square centered at the origin, each symmetry acts by a combination of negating and swapping the Cartesian coordinates of points.
The symmetries permute the eight isosceles triangles between the half-edges and the square's center (which stays in place); any of these triangles can be taken as the fundamental region of the transformations. Each two vertices, each two edges, and each two half-edges are mapped one to the other by at least one symmetry (exactly one for half-edges). All regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
s also have these properties, which are expressed by saying that symmetries of a square and, more generally, a regular polygon act transitively on vertices and edges, and simply transitively on half-edges.
Combining any two of these transformations by performing one after the other continues to take the square to itself, and therefore produces another symmetry. Repeated rotation produces another rotation with the summed rotation angle. Two reflections with the same axis return to the identity transformation, while two reflections with different axes rotate the square. A rotation followed by a reflection, or vice versa, produces a different reflection. This composition operation gives the eight symmetries of a square the mathematical structure of a group, called the ''group of the square'' or the '' dihedral group of order eight''.[ Other quadrilaterals, like the rectangle and rhombus, have only a ]subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
of these symmetries.
The shape of a square, but not its size, is preserved by similarities of the plane. Other kinds of transformations of the plane can take squares to other kinds of quadrilateral. An affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More general ...
can take a square to any parallelogram, or vice versa; a projective transformation can take a square to any convex quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
, or vice versa. This implies that, when viewed in perspective, a square can look like any convex quadrilateral, or vice versa. A Möbius transformation can take the vertices of a square (but not its edges) to the vertices of a harmonic quadrilateral.
The wallpaper groups are symmetry groups of two-dimensional repeating patterns. For many of these groups the basic unit of repetition (the unit cell of its period lattice) can be a square, and for three of these groups, p4, p4m, and p4g, it must be a square.
Inscribed and circumscribed circles
The inscribed circle of a square is the largest circle that can fit inside that square. Its center is the center point of the square, and its radius (the inradius
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
of the square) is . Because this circle touches all four sides of the square (at their midpoints), the square is a tangential quadrilateral. The circumscribed circle of a square passes through all four vertices, making the square a cyclic quadrilateral. Its radius, the circumradius, is . If the inscribed circle of a square has tangency points on , on , on , and on , then for any point on the inscribed circle, If is the distance from an arbitrary point in the plane to the vertex of a square and is the circumradius of the square, then
If and are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then and where is the circumradius of the square.
Applications
Squares are so well-established as the shape of tiles that the Latin
Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
word tessera, for a small tile as used in mosaic
A mosaic () is a pattern or image made of small regular or irregular pieces of colored stone, glass or ceramic, held in place by plaster/Mortar (masonry), mortar, and covering a surface. Mosaics are often used as floor and wall decoration, and ...
s, comes from an ancient Greek word for the number four, referring to the four corners of a square tile. Graph paper, preprinted with a square tiling, is widely used for data visualization using Cartesian coordinates. The pixel
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a Raster graphics, raster image, or the smallest addressable element in a dot matrix display device. In most digital display devices, p ...
s of bitmap images, as recorded by image scanner
An image scanner (often abbreviated to just scanner) is a device that optically scans images, printed text, handwriting, or an object and converts it to a digital image. The most common type of scanner used in the home and the office is the flatbe ...
s and digital camera
A digital camera, also called a digicam, is a camera that captures photographs in Digital data storage, digital memory. Most cameras produced today are digital, largely replacing those that capture images on photographic film or film stock. Dig ...
s or displayed on electronic visual displays, conventionally lie at the intersections of a square grid, and are often considered as small squares, arranged in a square tiling. Standard techniques for image compression and video compression
In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compression ...
, including the JPEG
JPEG ( , short for Joint Photographic Experts Group and sometimes retroactively referred to as JPEG 1) is a commonly used method of lossy compression for digital images, particularly for those images produced by digital photography. The degr ...
format, are based on the subdivision of images into larger square blocks of pixels. The quadtree data structure used in data compression and computational geometry is based on the recursive subdivision of squares into smaller squares.
Architectural structures from both ancient and modern cultures have featured a square floor plan, base, or footprint. Ancient examples include the Egyptian pyramids
The Egyptian pyramids are ancient masonry structures located in Egypt. Most were built as tombs for the pharaohs and their consorts during the Old Kingdom of Egypt, Old and Middle Kingdom of Egypt, Middle Kingdom periods. At least 138 identi ...
, Mesoamerican pyramids such as those at Teotihuacan
Teotihuacan (; Spanish language, Spanish: ''Teotihuacán'', ; ) is an ancient Mesoamerican city located in a sub-valley of the Valley of Mexico, which is located in the State of Mexico, northeast of modern-day Mexico City.
Teotihuacan is ...
, the Chogha Zanbil ziggurat in Iran, the four-fold design of Persian walled gardens, said to model the four rivers of Paradise,
and later structures inspired by their design such as the Taj Mahal
The Taj Mahal ( ; ; ) is an ivory-white marble mausoleum on the right bank of the river Yamuna in Agra, Uttar Pradesh, India. It was commissioned in 1631 by the fifth Mughal Empire, Mughal emperor, Shah Jahan () to house the tomb of his belo ...
in India, the square bases of Buddhist stupas, and East Asian pagodas, buildings that symbolically face to the four points of the compass and reach to the heavens. Norman keeps such as the Tower of London
The Tower of London, officially His Majesty's Royal Palace and Fortress of the Tower of London, is a historic citadel and castle on the north bank of the River Thames in central London, England. It lies within the London Borough of Tower Hamle ...
often take the form of a low square tower. In modern architecture, a majority of skyscraper
A skyscraper is a tall continuously habitable building having multiple floors. Most modern sources define skyscrapers as being at least or in height, though there is no universally accepted definition, other than being very tall high-rise bui ...
s feature a square plan for pragmatic rather than aesthetic or symbolic reasons.
The stylized nested squares of a Tibetan mandala, like the design of a stupa, function as a miniature model of the cosmos. Some formats for film photography use a square aspect ratio
The aspect ratio of a geometry, geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangl ...
, notably Polaroid cameras, medium format cameras, and Instamatic cameras. Painters known for their frequent use of square frames and forms include Josef Albers
Josef Albers ( , , ; March 19, 1888March 25, 1976) was a German-born American artist and Visual arts education, educator who is considered one of the most influential 20th-century art teachers in the United States. Born in 1888 in Bottrop, Westp ...
, and Piet Mondrian
Pieter Cornelis Mondriaan (; 7 March 1872 – 1 February 1944), known after 1911 as Piet Mondrian (, , ), was a Dutch Painting, painter and Theory of art, art theoretician who is regarded as one of the greatest artists of the 20th century. He w ...
.
Baseball diamonds and boxing ring
A boxing ring, often referred to simply as a ring or the squared circle, is the space in which a boxing match occurs. A modern ring consists of a square raised platform with a post at each corner. Four ropes are attached to the posts and pulled p ...
s are square despite being named for other shapes. In the quadrille and square dance, four couples form the sides of a square. In Samuel Beckett
Samuel Barclay Beckett (; 13 April 1906 – 22 December 1989) was an Irish writer of novels, plays, short stories, and poems. Writing in both English and French, his literary and theatrical work features bleak, impersonal, and Tragicomedy, tra ...
's minimalist television play '' Quad'', four actors walk along the sides and diagonals of a square.
The square go board is said to represent the earth, with the 361 crossings of its lines representing days of the year. The chessboard inherited its square shape from a pachisi-like Indian race game and in turn passed it on to checkers
Checkers (American English), also known as draughts (; English in the Commonwealth of Nations, Commonwealth English), is a group of Abstract strategy game, strategy board games for two players which involve forward movements of uniform game ...
. In two ancient games from Mesopotamia
Mesopotamia is a historical region of West Asia situated within the Tigris–Euphrates river system, in the northern part of the Fertile Crescent. Today, Mesopotamia is known as present-day Iraq and forms the eastern geographic boundary of ...
and Ancient Egypt
Ancient Egypt () was a cradle of civilization concentrated along the lower reaches of the Nile River in Northeast Africa. It emerged from prehistoric Egypt around 3150BC (according to conventional Egyptian chronology), when Upper and Lower E ...
, the Royal Game of Ur and Senet, the game board itself is not square, but rectangular, subdivided into a grid of squares. The ancient Greek Ostomachion puzzle (according to some interpretations) involves rearranging the pieces of a square cut into smaller polygons, as does the Chinese tangram. Another set of puzzle pieces, the polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
Polyominoes have been used in popu ...
s, are formed from squares glued edge-to-edge. Medieval and Renaissance horoscopes were arranged in a square format, across Europe, the Middle East, and China. Other recreational uses of squares include the shape of origami paper, and a common style of quilting involving the use of square quilt blocks.
Squares are a common element of graphic design
Graphic design is a profession, academic discipline and applied art that involves creating visual communications intended to transmit specific messages to social groups, with specific objectives. Graphic design is an interdisciplinary branch of ...
, used to give a sense of stability, symmetry, and order. In heraldry
Heraldry is a discipline relating to the design, display and study of armorial bearings (known as armory), as well as related disciplines, such as vexillology, together with the study of ceremony, Imperial, royal and noble ranks, rank and genealo ...
, a canton (a design element in the top left of a shield) is normally square, and a square flag is called a banner. The flag of Switzerland is square, as are the flags of the Swiss cantons. QR codes are square and feature prominent nested square alignment marks in three corners. Robertson screws have a square drive socket. Crackers and sliced cheese
Cheese is a type of dairy product produced in a range of flavors, textures, and forms by coagulation of the milk protein casein. It comprises proteins and fat from milk (usually the milk of cows, buffalo, goats or sheep). During prod ...
are often square, as are waffles. Square foods named for their square shapes include caramel squares, date squares, lemon squares, square sausage, and Carré de l'Est cheese.
Constructions
Coordinates and equations
A unit square
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and .
Cartesian coordinates
In a Cartesian coordinat ...
is a square of side length one. Often it is represented in Cartesian coordinates as the square enclosing the points that have and . Its vertices are the four points that have 0 or 1 in each of their coordinates.
An axis-parallel square with its center at the point and sides of length (where is the inradius, half the side length) has vertices at the four points . Its interior consists of the points with , and its boundary consists of the points with .
A diagonal square with its center at the point and diagonal of length (where is the circumradius, half the diagonal) has vertices at the four points and . Its interior consists of the points with