Definitions

Square (geometry)

Square

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

In Euclidean Geometry geometry, a square is a regular polygon with four equal sides. In Euclidean geometry, it has four 90 degree angles. A square with vertices ABCD would be denoted .

Classification

A square (regular quadrilateral) is a special case of a rectangle as it has four right angles and equal parallel sides. Likewise it is also a special case of a rhombus, kite, parallelogram, and trapezoid.

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.

Properties

Each angle in a square is equal to 90 degrees, or a right angle.

The diagonals of a square are equal. Conversely, if the diagonals of a rhombus are equal, then that rhombus must be a square. The diagonals of a square are $sqrt\left\{2\right\}$ (about 1.41) times the length of a side of the square. This value, known as Pythagoras’ constant, was the first number proven to be irrational.

If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths) then it is a square.

Other facts

• It has all equal sides and the angles add up to 360 degrees.
• 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 ().
• 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}.