Square

## Characterizations

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

* 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 :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]

* The diagonals of a square bisect each other 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 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

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 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.[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

* 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 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.
* 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 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 D4.
* 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]

## Coordinates and equations

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 (*x*i, *y*i) with −1 < *xi* < 1 and −1 < *yi* < 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 also be used to describe the boundary of a square with center coordinates (*a*, *b*) and a horizontal or vertical radius of *r*.