Square

Square
cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

In

ABCD would be denoted ${\displaystyle \square }$ ABCD.[1]

Characterizations

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 ${\displaystyle A={\tfrac {1}{2}}(a^{2}+c^{2})={\tfrac {1}{2}}(b^{2}+d^{2}).}$^{[4]: Corollary 15}

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 (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-
  orthoplexes
  .

Perimeter and area

The perimeter of a square whose four sides have length ${\displaystyle \ell }$ is

${\displaystyle P=4\ell }$

and the area A is

${\displaystyle A=\ell ^{2}.}$^[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

to mean raising to the second power.

The area can also be calculated using the diagonal d according to

${\displaystyle A={\frac {d^{2}}{2}}.}$

In terms of the

circumradius

R, the area of a square is

${\displaystyle A=2R^{2};}$

since the area of the circle is ${\displaystyle \pi R^{2},}$ the square fills ${\displaystyle 2/\pi \approx 0.6366}$ of its circumscribed circle.

In terms of the

r, the area of the square is

${\displaystyle A=4r^{2};}$

hence the area of the

is ${\displaystyle \pi /4\approx 0.7854}$ 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.^[6] Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds:

${\displaystyle 16A\leq P^{2}}$

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

Other facts

• The diagonals of a square are ${\displaystyle {\sqrt {2}}}$ (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
  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 D₄.
• 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]

${\displaystyle 2(PH^{2}-PE^{2})=PD^{2}-PB^{2}.}$

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

${\displaystyle {\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+3R^{4}=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+R^{2}\right)^{2}.}$

• If ${\displaystyle L}$ and ${\displaystyle d_{i}}$ are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then [10]

${\displaystyle d_{1}^{2}+d_{3}^{2}=d_{2}^{2}+d_{4}^{2}=2(R^{2}+L^{2})}$

and

${\displaystyle d_{1}^{2}d_{3}^{2}+d_{2}^{2}d_{4}^{2}=2(R^{4}+L^{4}),}$

where ${\displaystyle R}$ 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 −1 < x_i < 1 and −1 < y_i < 1. The equation

${\displaystyle \max(x^{2},y^{2})=1}$

specifies the boundary of this square. This equation means "x² or y², 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 ${\displaystyle {\sqrt {2}}.}$ Then the circumcircle has the equation

${\displaystyle x^{2}+y^{2}=2.}$

Alternatively the equation

${\displaystyle \left|x-a\right|+\left|y-b\right|=r.}$

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.

Construction

The following animations show how to construct a square using a

.

Symmetry

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:

• 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

.

Squares inscribed in triangles

Every

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

.

In 1882, the task was proven to be impossible as a consequence of the

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.

Examples:

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}.

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.

Crossed square

A crossed square is a

with a common vertex, but the geometric intersection is not considered a vertex.

A crossed square is sometimes likened to a

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

.

Graphs

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

• Cube
• Pythagorean theorem
• Square lattice
• Square number
• Square root
• Squaring the square
• Squircle
• Unit square

References

External links

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

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: List of regular polytopes and compounds