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Regular polygon 4 annotated.svg
A regular quadrilateral
Type Regular polygon
Edges and vertices 4
Schläfli symbol {4}
Coxeter–Dynkin diagrams CDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 2.pngCDel node 1.png
Symmetry group Dihedral (D4), order 2×4
Internal angle (degrees)90°
Properties Convex, cyclic, equilateral, isogonal, isotoxal

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ABCD. [1]



A convex quadrilateral is a square if and only if it is any one of the following: [2] [3]


A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a trapezoid (one pair of opposite sides parallel), a parallelogram (all opposite sides parallel), a quadrilateral or tetragon (four-sided polygon), and a rectangle (opposite sides equal, right-angles), and therefore has all the properties of all these shapes, namely: [5]

Perimeter and area

The area of a square is the product of the length of its sides. Five Squared.svg
The area of a square is the product of the length of its sides.

The perimeter of a square whose four sides have length is

and the area A is


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

In terms of the circumradius R, the area of a square is

since the area of the circle is the square fills of its circumscribed circle.

In terms of the inradius r, the area of the square is

hence the area of the inscribed circle is 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. [6] 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.

Other facts

where is the circumradius of the square.

Coordinates and equations

{\displaystyle |x|+|y|=2}
plotted on Cartesian coordinates. Square equation plot.svg
plotted on Cartesian coordinates .

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 (xi, yi) with −1 < xi < 1 and −1 < yi < 1. The equation

specifies the boundary of this square. This equation means "x2 or y2, whichever is larger, equals 1." The circumradius of this square (the radius of a circle drawn through the square's vertices) is half the square's diagonal, and is equal to Then the circumcircle has the equation

Alternatively the equation

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


The following animations show how to construct a square using a compass and straightedge. This is possible as 4 = 22, a power of two.

Square at a given circumcircle Straight Square Inscribed in a Circle 240px.gif
Square at a given circumcircle
Square at a given side length,
right angle by using Thales' theorem
Square at a given diagonal


The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the square is r8 and no symmetry is labeled a1. Symmetries of square.svg
The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the square is r8 and no symmetry is labeled a1.

The square has Dih4 symmetry, order 8. There are 2 dihedral subgroups: Dih2, Dih1, and 3 cyclic subgroups: Z4, Z2, and Z1.

A square is a special case of many lower symmetry quadrilaterals:

These 6 symmetries express 8 distinct symmetries on a square. John Conway labels these by a letter and group order. [11]

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 rectangle, 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 parallelogram.

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 has three inscribed 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 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, proposed by ancient geometers, is the problem of constructing a square with the same area as a given circle, by using only a finite number of steps with compass and straightedge.

In 1882, the task was proven to be impossible as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (π) is a transcendental number rather than an algebraic irrational number; that is, it is not the root of any polynomial 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, a square is a polygon whose edges are great circle 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.


Tetragonal dihedron.png
Two squares can tile the sphere with 2 squares around each vertex and 180-degree internal angles. Each square covers an entire hemisphere and their vertices lie along a great circle. This is called a spherical square dihedron. The Schläfli symbol is {4,2}.
Square on sphere.svg
Six squares can tile the sphere with 3 squares around each vertex and 120-degree internal angles. This is called a spherical cube. The Schläfli symbol is {4,3}.
Square on hyperbolic plane.png
Squares can tile the hyperbolic plane with 5 around each vertex, with each square having 72-degree internal angles. The Schläfli symbol is  {4,5}. In fact, for any n ≥ 5 there is a hyperbolic tiling with n squares about each vertex.

Crossed square

Crossed-square Tetrahemihexahedron vertfig.png

A crossed square is a faceting 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. It appears as two 45-45-90 triangle 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. the crossed rectangle is related, as a faceting of the rectangle, both special cases of crossed quadrilaterals. [12]

The interior of a crossed square can have a polygon density 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:

It exists in the vertex figure of a uniform star polyhedra, the tetrahemihexahedron.


3-simplex (3D) Tetrahedron petrie.png
3-simplex (3D)

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 of the 4 vertices and 6 edges of the regular 3-simplex (tetrahedron).

See also

Related Research Articles

Area Size of a two-dimensional surface

Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve or the volume of a solid.

Quadrilateral Polygon with four sides and four corners

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". Another name for it is tetragon, derived from greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to e.g., pentagon. "Gon" being "angle" also is at the root of calling it quadrangle, 4-angle, in analogy to triangle. A quadrilateral with vertices , , and is sometimes denoted as .

Hexagon Shape with six sides

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°.

Rectangle Quadrilateral with four right angles

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 ; or 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. A rectangle with vertices ABCD would be denoted as  ABCD.

Kite (geometry)

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.

Parallelogram Quadrilateral with two pairs of parallel sides

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.

Rhombus Quadrilateral in which all sides have the same length

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.

Cyclic quadrilateral Quadrilateral whose vertices can all fall on a single circle

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.

Trapezoid Convex quadrilateral with at least one pair of parallel sides

In English outside North America, a convex quadrilateral in Euclidean geometry, with at least one pair of parallel sides, is referred to as a trapezium ; in American and Canadian English this is usually referred to as a trapezoid. The parallel sides are called the bases of the trapezoid. The other two sides are called the legs if they are not parallel; otherwise, the trapezoid is a parallelogram, and there are two pairs of bases). A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast with the special cases below.

Thaless theorem Angle formed by a point on a circle and the 2 ends of a diameter is a right angle

In geometry, Thales' 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, but it is sometimes attributed to Pythagoras.

Octagon Polygon shape with eight sides

In geometry, an octagon is an eight-sided polygon or 8-gon.

Decagon Shape with ten sides

In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.

Midpoint Point on a line segment which is equidistant from both endpoints

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.

Isosceles trapezoid Trapezoid symmetrical about an axis

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 equal 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.

Antiparallelogram Polygon with four crossed edges of two lengths

In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel. Instead, sides in the longer pair cross each other as in a scissors mechanism. Antiparallelograms are also called contraparallelograms or crossed parallelograms.

Varignons theorem The midpoints of the sides of an arbitrary quadrilateral form a parallelogram

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.

Pentagon Shape with five sides

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

Orthodiagonal quadrilateral

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.

Ex-tangential quadrilateral

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.

Equidiagonal 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.


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  3. "Problem Set 1.3". jwilson.coe.uga.edu. Retrieved 2017-12-12.
  4. Josefsson, Martin, "Properties of equidiagonal quadrilaterals" Forum Geometricorum, 14 (2014), 129-144.
  5. "Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram". www.mathsisfun.com. Retrieved 2020-09-02.
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  10. Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids". Communications in Mathematics and Applications. 11: 335–355. arXiv: 2010.12340 .
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  12. Wells, Christopher J. "Quadrilaterals". www.technologyuk.net. Retrieved 2017-12-12.
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds