# Square

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A square is an equilateral (equally-lengthed sides) and equiangular quadrilateral.

A square is both a rhombus and a rectangle, simultaneously. Thus, a square shares the properties of each:

• All four sides are of equal length (congruent)
• All four corner angles are of equal measure (congruent)
• All four corner angles are right angles
• Adjacent sides meet at right angles
• Opposite sides are parallel to one another
• Parallelogram
• Diagonals bisect the angles of the corners they connect at one-half a right angle
• Diagonals bisect one another
• Diagonals are of equal length
• Diagonals intersect at right angles
Square

A square is a regular quadrilateral.
Edges and vertices4
Schläfli symbols{4}
t{2} or {}x{}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D4)
Area
(with t=edge length)
t2
Internal angle
(degrees)
90°

In Euclidean geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles (90 degree angles, or right angles). A square with vertices ABCD would be denoted Template:Squarenotation.

## Classification

Two-dimensional object made up with four points, and four equal line-segments.

## The mensuration formula

The perimeter of a square whose sides have length t is

$P=4t.$

And the area is

$A=t^2.$

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.

## Standard coordinates

The coordinates for the vertices of a square centered at the origin and with side length 2 are (±1, ±1), while the interior of the same consists of all points (x0, x1) with −1 < xi < 1.

## Equations

The equation max$(x^2, y^2) = 1$ describes a square. This means "$x^2$ or $y^2$, whichever is larger, equals 1." The circumradius of this square is $\sqrt{2}$.

## Properties

The diagonals of a square bisect each other.

The diagonals of a square bisect its angles.

The diagonals of a square are perpendicular.

Opposite sides of a square are both parallel and equal.

All four angles of a square are equal. (Each is $\frac{360}{4} = 90$ degrees, so every angle of a square is a right angle.)

The diagonals of a square are equal.

## Other facts

• If the diagonals of a rhombus are equal, then that rhombus must be a square. The diagonals of a square are $\sqrt{2}$ (about 1.414) times the length of a side of the square. This value, known as Pythagoras’ constant, was the first number proven to be irrational.
• A square can also be defined as a rectangle with all sides equal, or a rhombus with all angles equal, or 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. (Rectangle (four equal angles) + Rhombus (four equal sides) = Square)
• If a circle is circumscribed around a square, the area of the circle is $\pi/2$ (about 1.57) times the area of the square.
• If a circle is inscribed in the square, the area of the circle is $\pi/4$ (about 0.79) times the area of the square.
• A square has a larger area than any other quadrilateral with the same perimeter ([1]).
• 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 (in Goldman geometry). There are four lines of reflectional symmetry and it has rotational symmetry through 90°, 180° and 270°. Its symmetry group is the dihedral group $D_4$.

## 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.

In hyperbolic geometry, squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger squares have smaller angles.

Examples:

 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 degrees. 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}.