This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2010) Square A square is a regular quadrilateral. Families 2-hypercube, 2-orthoplex Edges and vertices 4 Schläfli symbol {4} Coxeter–Dynkin diagrams 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 ABCD. Contents 1 Perimeter and area 2 Standard coordinates 3 Equations 4 Properties 5 Other facts 6 Non-Euclidean geometry 7 Graphs 8 See also 9 External links 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 and the area 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. 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. Construction of a square using a compass and straightedge. 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 . 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 and meet at 90 degrees. 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 (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. 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: 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}. 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 Hypercubes family Point (geometry) - - Line (geometry) - {} Square - {4} Cube - {4,3} Tesseract - {4,3,3} Penteract - {4,3,3,3} Hexeract - {4,3,3,3,3} Hepteract - {4,3,3,3,3,3} Octeract - {4,3,3,3,3,3,3} Enneract - {4,3,3,3,3,3,3,3} Dekeract - {4,3,3,3,3,3,3,3,3} Hendekeract - {4,3,3,3,3,3,3,3,3,3} Dodekeract - {4,3,3,3,3,3,3,3,3,3,3} n-eract External links Wikimedia Commons has media related to: Squares (geometry) 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 v · d · eRegular polygons Listed by number of sides 1–10 sides Henagon (Monogon) · Digon · Triangle (Trigon) · Quadrilateral (Tetragon) · Pentagon · Hexagon · Heptagon · Octagon · Nonagon (Enneagon) · Decagon 11–20 sides Hendecagon · Dodecagon · Triskaidecagon · Tetradecagon · Pentadecagon · Hexadecagon · Heptadecagon · Octadecagon · Nonadecagon (Enneadecagon) · Icosagon Others Apeirogon Star polygons Pentagram · Hexagram · Heptagram · Octagram · Enneagram · Decagram · Hendecagram · Dodecagram