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