Square (geometry)

Square
A square is a regular quadrilateral.
Type	Regular polygon
Edges and vertices	4
Schläfli symbol	{4}
Coxeter–Dynkin diagram
Symmetry group	Dihedral (D4)
Area	t2 (with t = edge length)
Internal angle (degrees)	90°
Dual polygon	dual polygon of this shape
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In 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 $\square$ ABCD

The square belong to the families of 2-hypercube and 2-orthoplex.

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 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 rhombus with equal diagonals

Perimeter and area

The area of a square is the product of the length of its sides.

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 < x_i < 1.

Construction of a square using a compass and straightedge.

Equations

The equation max(x²,y²) = 1 describes a square of side = 2, centered at the origin. 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 $\scriptstyle \sqrt{2}$.

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 other and meet at 90°
• 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 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.)
• The diagonals of a square are equal.

Other facts

• The diagonals of a square are $\scriptstyle \sqrt{2}$ (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 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 π / 2 (about 1.571) times the area of the square.
• If a circle is inscribed in the square, the area of the circle is π / 4 (about 0.7854) 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. 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.

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°. 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}.

Graphs

The K₄ complete graph is often drawn as a square with all 6 edges connected. This graph also represents an orthographic projection of the 4 vertices and 6 edges of the regular 3-simplex (tetrahedron).

3-simplex (3D)

See also

• Cube
• Pythagorean theorem
• Square lattice
• Unit square
• Squircle

External links

• Animated course (Construction, Circumference, Area)
• Weisstein, Eric W., "Square" from MathWorld.
• Definition and properties of a square With interactive applet
• Animated applet illustrating the area of a square
• www.mathisfun.com/quadrilaterals.html Everything you want to know about quadrilaterals.