In geometry, a **square** is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or right angles). It can additionally be defined as a rectangle in which two adjacent sides have equal length. A square with vertices *ABCD* would be denoted *ABCD*.

## Properties

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

- The diagonals of a square bisect each additional and meet at 90°
- The diagonals of a square bisect its angles.
- Opposite sides of a square are both parallel and equal in length.
- All four angles of a square are equal. (Each is 360°/4 = 90°, so every angle of a square is a right angle.)
- All four sides of a square are equal.
- The diagonals of a square are equal.
- The square is the n=2 case of the families of n-hypercubes and n-orthoplexes.
- A square has Schläfli symbol {4}. A truncated square, t{4}, is an octagon, {8}. An alternated square, h{4}, is a digon, {2}.

### Perimeter and area

The perimeter of a square whose four sides have length is

and the area *A* is

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 additionally 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 approximately 0.6366 of its circumscribed circle.

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

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 holds:

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

A convex quadrilateral with successive sides *a, b, c, d* is a square if and only if ^{:Corollary 15}

### Other facts

- 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, was the first number proven to be irrational.
- A square can additionally 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.
- If a circle is circumscribed around a square, the area of the circle is (about 1.5708) times the area of the square.
- If a circle is inscribed in the square, the area of the circle is (about 0.7854) times the area of the square.
- A square has a larger area than any additional quadrilateral with the same perimeter.
- 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}. - 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,

- 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

## Coordinates and equations

The coordinates for the vertices of a square with vertical and horizontal sides, centred 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 equals . Then the circumcircle has the equation

Alternatively the equation

can additionally be used to describe the boundary of a square with centre coordinates (*a*, *b*) and a horizontal or vertical radius of *r*.

## Construction

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.

## Symmetry

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 a large number of lower symmetry convex 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 quadrilateral where the diagonals are equal and are the perpendicular bisectors of each other, i.e. 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.

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 additional 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-angled 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's filled by the square is no more than 1/2.

## Squaring the circle

Squaring the circle is the problem, proposed by ancient geometers, 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 isn't 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 don't 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 Euclidean plane with 4 around each vertex, with each square having an internal angle of 90°. The Schläfli symbol is {4,4}. | 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's a hyperbolic tiling with n squares about each vertex. |

## Crossed square

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 triangle with a common vertex, but the geometric intersection isn't considered a vertex.

A crossed square is at times likened to a bow tie or butterfly. the crossed rectangle is related, as a faceting of the rectangle, both special cases of crossed quadrilaterals.

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.

## Graphs

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 additionally represents an orthographic projection of the 4 vertices and 6 edges of the regular 3-simplex (tetrahedron).