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Square
Square (geometry).svg
A square is a regular quadrilateral.
Edges and vertices 4
Schläfli symbol {4}
Coxeter–Dynkin diagrams CDW ring.pngCDW 4.pngCDW dot.png
Symmetry group Dihedral (D4)
Area
(with t=edge length)
t2
Internal angle
(degrees)
90°
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.

Contents

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 < xi < 1.

Equations

The equation max(x2,y2) = 1 describes a square. This means "x2 or y2, whichever is larger, equals 1." The circumradius of this square is \sqrt{2}.

Properties

A square is both a rhombus (equal sides) and a rectangle (equal angles) and therefore has all the properties of both these shapes, namely:

  • 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 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 π / 2 (about 1.57) times the area of the square.
  • If a circle is inscribed in the square, the area of the circle is π / 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 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:

Square on sphere.svg
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}.
Square on plane.svg
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}.
Square on hyperbolic plane.png
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}.

See also

External links

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Simple English

[[File:|right|thumb|A square and its diagonals.]]

A square is a shape with four equal sides and four corners that are all right angles (90 degrees). The diagonals of a square also cross at right angles. The angle between any diagonal and a side of a square is 45 degrees. A square has rotational symmetry of four. It has four lines of regular symmetry.

A square is a type of rectangle with all sides of equal length.


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