In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal

angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

s (90-

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

angles, π/2 radian angles, or right angles). It can also be defined as a

rectangle 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 containi ...

with two equal-length adjacent sides. It is the only regular polygon whose internal angle,

central angle A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc le ...

, and external angle are all equal (90°), and whose

diagonals 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 δ� ...

are all equal in length. A square with vertices ''ABCD'' would be denoted .

Characterizations

A convex quadrilateral is a square

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

it is any one of the following: * A

with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A

parallelogram 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 equa ...

with one right vertex angle and two adjacent equal sides * A quadrilateral 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 * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' whose area is

A= \tfrac(a^2+c^2)=\tfrac(b^2+d^2).

Josefsson, Martin
"Properties of equidiagonal quadrilaterals"
''Forum Geometricorum'', 14 (2014), 129-144.

Properties

A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a

trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a Convex polygon, convex quadri ...

(one pair of opposite sides parallel), a

(all opposite sides parallel), a quadrilateral or tetragon (four-sided polygon), and 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 360°/4 = 90°, a right angle). * The central angle of a square is equal to 90° (360°/4). * The external angle of a square is equal to 90°. * The diagonals of a square are equal and

bisect Bisect, or similar, may refer to: Mathematics * Bisection, in geometry, dividing something into two equal parts * Bisection method, a root-finding algorithm * Equidistant set Other uses * Bisect (philately), the use of postage stamp halves * Bis ...

each other, meeting at 90°. * The diagonal of a square bisects its internal angle, forming

adjacent angles In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles are ...

of 45°. * All four sides of a square are equal. * Opposite sides of a square are

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM ...

. * The square is the n=2 case of the families of n- hypercubes and n- orthoplexes. * A square has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

. A truncated square, t, is an

octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...

, . An alternated square, h, is a digon, .

Perimeter and area

The perimeter of a square whose four sides have length

\ell

is :

P=4\ell

and the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

''A'' is :

A=\ell^2.

Since four squared equals sixteen, a four by four square has an area equal to its perimeter. The only other quadrilateral with such a property is that of a three by six rectangle. In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term '' square'' to mean raising to the second power. The area can also be calculated using the diagonal ''d'' according to :

A=\frac.

In terms of the

circumradius In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

''R'', the area of a square is :

A=2R^2;

since the area of the circle is

\pi R^2,

the square fills

2/\pi \approx 0.6366

of its circumscribed circle. In terms of 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. ...

''r'', the area of the square is :

A=4r^2;

hence the area of the

inscribed circle 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. ...

\pi/4 \approx 0.7854

of that of the square. Because it is a regular polygon, 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 In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...

holds: :

16A\le P^2

with equality if and only if the quadrilateral is a square.

Other facts

* The diagonals of a square are

\sqrt

(about 1.414) times the length of a side of the square. This value, known as the

square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...

or Pythagoras' constant, was the first number proven to be irrational. * A square can also be defined as a

with equal diagonals that bisect the angles. * If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square. * A square has a larger area than any other quadrilateral with the same perimeter. * A square tiling is one of three

regular tilings This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ' ...

of the plane (the others are the equilateral triangle and the

regular hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

). * The square is in two families of polytopes in two dimensions:

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...

and the

cross-polytope In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...

. The

for the square is . * The square is a highly symmetric object. There are four lines of

reflectional symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...

and it has

rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...

of order 4 (through 90°, 180° and 270°). Its

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...

is the dihedral group D₄. * A square can be

inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figur ...

inside any regular polygon. The only other polygon with this property is the equilateral triangle. * If the inscribed circle of a square ''ABCD'' has tangency points ''E'' on ''AB'', ''F'' on ''BC'', ''G'' on ''CD'', and ''H'' on ''DA'', then for any point ''P'' on the inscribed circle, ::

2(PH^2-PE^2) = PD^2-PB^2.

* If

d_i

is the distance from an arbitrary point in the plane to the ''i''-th vertex of a square and

R

is the

of the square, then ::

\frac + 3R^4 = \left(\frac + R^2\right)^2.

* If

L

and

d_i

are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then ::

d_1^2 + d_3^2 = d_2^2 + d_4^2 = 2(R^2+L^2)

:and ::

d_1^2d_3^2 + d_2^2d_4^2 = 2(R^4+L^4),

:where

R

is the circumradius of the square.

Coordinates and equations

The coordinates for the vertices of a square with vertical and horizontal sides, centered at the origin and with side length 2 are (±1, ±1), while the interior of this square consists of all points (''x''_i, ''y''_i) with and . The equation :

\max(x^2, y^2) = 1

specifies the boundary of this square. This equation means "''x''² or ''y''², whichever is larger, equals 1." The

of this square (the radius of a circle drawn through the square's vertices) is half the square's diagonal, and is equal to

\sqrt.

Then the circumcircle has the equation :

x^2 + y^2 = 2.

Alternatively the equation :

\left, x - a\ + \left, y - b\ = r.

can also be used to describe the boundary of a square with center

coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...

(''a'', ''b''), and a horizontal or vertical radius of ''r''. The square is therefore the shape of a topological ball according to the L₁ distance metric.

Construction

The following animations show how to construct a square using a

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 ideali ...

. This is possible as 4 = 2², a

power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative ...

Straight Square Inscribed in a Circle 240px

Symmetry

The ''square'' has Dih₄ symmetry,

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

8. There are 2 dihedral subgroups: Dih₂, Dih₁, and 3

cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in soc ...

subgroups: Z₄, Z₂, and Z₁. A square is a special case of many lower symmetry quadrilaterals: * A rectangle with two adjacent equal sides * A quadrilateral with four equal sides and four

s * A parallelogram with one right angle and two adjacent equal sides * A rhombus with a right angle * A rhombus with all angles equal * A rhombus with equal diagonals These 6 symmetries express 8 distinct symmetries on a square.

John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches o ...

labels these by a letter and group order. Each subgroup symmetry allows one or more degrees of freedom for irregular quadrilaterals. r8 is full symmetry of the square, and a1 is no symmetry. d4 is the symmetry of a

, and p4 is the symmetry of a rhombus. These two forms are duals of each other, and have half the symmetry order of the square. d2 is the symmetry of an isosceles trapezoid, and p2 is the symmetry of a kite. g2 defines the geometry of a

. Only the g4 subgroup has no degrees of freedom, but can seen as a square with directed edges.

Squares inscribed in triangles

Every

acute triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...

has three

squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two ''distinct'' inscribed squares. An

obtuse triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...

has only one inscribed square, with a side coinciding with part of the triangle's longest side. The fraction of the triangle's area that is filled by the square is no more than 1/2.

Squaring the circle

Squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty ...

, proposed by ancient geometers, is the problem of constructing a square with the same area as a given

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

, by using only a finite number of steps with

. In 1882, the task was proven to be impossible as a consequence of the Lindemann–Weierstrass theorem, which proves that pi () is a

transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...

rather than an algebraic irrational number; that is, it is not the

root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...

of any

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

with rational coefficients.

Non-Euclidean geometry

In non-Euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles. In

spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...

, a square is a polygon whose edges are

great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...

arcs of equal distance, which meet at equal angles. Unlike the square of plane geometry, the angles of such a square are larger than a right angle. Larger spherical squares have larger angles. In hyperbolic geometry, squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger hyperbolic squares have smaller angles. Examples:

Crossed square

A crossed square is a

faceting Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new Vertex (geometry), vertices. New edges of a faceted pol ...

of the square, a self-intersecting polygon created by removing two opposite edges of a square and reconnecting by its two diagonals. It has half the symmetry of the square, Dih₂, order 4. It has the same vertex arrangement as the square, and is

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

. It appears as two

45-45-90 triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45° ...

with a common vertex, but the geometric intersection is not considered a vertex. A crossed square is sometimes likened to a bow tie or

butterfly Butterflies are insects in the macrolepidopteran clade Rhopalocera from the Order (biology), order Lepidoptera, which also includes moths. Adult butterflies have large, often brightly coloured wings, and conspicuous, fluttering flight. The ...

. the

crossed rectangle 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 containin ...

is related, as a faceting of the rectangle, both special cases of

crossed quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

s. The interior of a crossed square can have a

polygon density In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions, representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It ...

of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise. A square and a crossed square have 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°). It exists in the

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

of a

uniform star polyhedra In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...

, the

tetrahemihexahedron In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin dia ...

Graphs

The K₄ complete graph is often drawn as a square with all 6 possible edges connected, hence appearing as a square with both diagonals drawn. This graph also represents an

orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...

of the 4 vertices and 6 edges of the regular 3-

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

(

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

References

External links

Animated course (Construction, Circumference, Area)

With interactive applet

{{Authority control Elementary shapes Types of quadrilaterals 4 (number) Constructible polygons