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 (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 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.