Square | |
---|---|

Type | Regular polygon |

Edges and vertices | 4 |

Schläfli symbol | {4} |

Coxeter–Dynkin diagrams | |

Symmetry group | Dihedral (D_{4}), order 2×4 |

Internal angle (degrees) | 90° |

Properties | Convex, cyclic, equilateral, isogonal, isotoxal |

Dual polygon | Self |

In Euclidean geometry, a **square** is a regular quadrilateral, which means that it has four sides of equal length 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 quadrilateral is a square if and only if it is any one of the following:^{ [2] }^{ [3] }

- A rectangle with two adjacent equal sides
- A rhombus with a right vertex angle
- A rhombus with all angles equal
- A parallelogram with one right vertex angle and two adjacent equal sides
- A quadrilateral with four equal sides and four right angles
- 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^{ [4] }^{: Corollary 15 }

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] }

- 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 each other, meeting at 90°.
- The diagonal of a square bisects its internal angle, forming adjacent angles of 45°.
- All four sides of a square are equal.
- Opposite sides of a square are parallel.

A square has Schläfli symbol {4}. A truncated square, t{4}, is an octagon, {8}. An alternated square, h{4}, is a digon, {2}. The square is the *n* = 2 case of the families of *n*-hypercubes and *n*-orthoplexes.

The perimeter of a square whose four sides have length is

and the area *A* is

^{ [1] }

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.

- The diagonals of a square are (about 1.414) times the length of a side of the square. This value, known as the square root of 2 or Pythagoras' constant,
^{ [1] }was the first number proven to be irrational. - A square can also be defined as a parallelogram 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.
^{ [7] } - A square tiling is one of three regular tilings of the plane (the others are the equilateral triangle and the regular hexagon).
- The square is in two families of polytopes in two dimensions: hypercube and the cross-polytope. The Schläfli symbol for the square is {4}.
- The square is a highly symmetric object. There are four lines of reflectional symmetry and it has rotational symmetry of order 4 (through 90°, 180° and 270°). Its symmetry group is the dihedral group D
_{4}. - A square can be inscribed 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,^{ [8] }

- If is the distance from an arbitrary point in the plane to the
*i*-th vertex of a square and is the circumradius of the square, then^{ [9] }

- If and are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then
^{ [10] }

- and
- where is the circumradius of the square.

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 −1 < *x*_{i} < 1 and −1 < *y*_{i} < 1. The equation

specifies the boundary of this square. This equation means "*x*^{2} or *y*^{2}, 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 L_{1} distance metric.

The following animations show how to construct a square using a compass and straightedge. This is possible as 4 = 2^{2}, a power of two.

The *square* has Dih_{4} symmetry, order 8. There are 2 dihedral subgroups: Dih_{2}, Dih_{1}, and 3 cyclic subgroups: Z_{4}, Z_{2}, and Z_{1}.

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 right angles
- 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 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 be seen as a square with directed edges.

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, 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.

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.

**Examples:**

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}. | 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}. | 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. |

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, Dih_{2}, order 4. It has the same vertex arrangement as the square, and is vertex-transitive. It appears as two 45-45-90 triangles 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:

- 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 of a uniform star polyhedra, the tetrahemihexahedron.

The K_{4} 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).

**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-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 analogue of the length of a curve or the volume of a solid . Two different regions may have the same area ; by synecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area".

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**, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a **quadrangle**, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .

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

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 used to refer to a non-square rectangle. A rectangle with vertices *ABCD* would be denoted as *ABCD*.

In Euclidean geometry, a **kite **is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as **deltoids**, but the word *deltoid* may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a **dart**, particularly if it is not convex.

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 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 geometry, a **trapezoid** in North American English, or **trapezium** in British English, is a quadrilateral that has at least one pair of parallel sides.

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

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°.

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 equal measure, or as a trapezoid whose diagonals have equal length. 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, and the diagonals have equal length. The base angles of an isosceles trapezoid are equal in measure.

In geometry, a **convex polygon** is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon. Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points.

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, each pair of sides is antiparallel with respect to the other, with sides in the longer pair crossing each other as in a scissors mechanism. Whereas a parallelogram's opposite angles are equal and oriented the same way, an antiparallelogram's are equal but oppositely oriented. Antiparallelograms are also called **contraparallelograms** or **crossed parallelograms**.

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

The **inscribed square problem**, also known as the **square peg problem** or the **Toeplitz' conjecture**, is an unsolved question in geometry: *Does every plane simple closed curve contain all four vertices of some square?* This is true if the curve is convex or piecewise smooth and in other special cases. The problem was proposed by Otto Toeplitz in 1911. Some early positive results were obtained by Arnold Emch and Lev Schnirelmann. As of 2020, the general case remains open.

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.

- 1 2 3 Weisstein, Eric W. "Square".
*Wolfram MathWorld*. Retrieved 2020-09-02. - ↑ Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 59, ISBN 1-59311-695-0.
- ↑ "Problem Set 1.3".
*jwilson.coe.uga.edu*. Retrieved 2017-12-12. - ↑ Josefsson, Martin, "Properties of equidiagonal quadrilaterals"
*Forum Geometricorum*, 14 (2014), 129–144. - ↑ "Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram".
*www.mathsisfun.com*. Retrieved 2020-09-02. - ↑ Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 in
*Mathematical Plums*(R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147. - ↑ Lundsgaard Hansen, Martin. "Vagn Lundsgaard Hansen".
*www2.mat.dtu.dk*. Retrieved 2017-12-12. - ↑ "Geometry classes, Problem 331. Square, Point on the Inscribed Circle, Tangency Points. Math teacher Master Degree. College, SAT Prep. Elearning, Online math tutor, LMS".
*gogeometry.com*. Retrieved 2017-12-12. - ↑ Park, Poo-Sung. "Regular polytope distances", Forum Geometricorum 16, 2016, 227–232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
- ↑ Meskhishvili, Mamuka (2021). "Cyclic Averages of Regular Polygonal Distances" (PDF).
*International Journal of Geometry*.**10**: 58–65. - ↑ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)
- ↑ Wells, Christopher J. "Quadrilaterals".
*www.technologyuk.net*. Retrieved 2017-12-12.

Wikimedia Commons has media related to Square (geometry) .

- Animated course (Construction, Circumference, Area)
- Definition and properties of a square With interactive applet
- Animated applet illustrating the area of a square

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