Square
A square The sides of a square and its diagonals meet at right angles.
Edges and vertices	4
Schläfli symbols	{4} {}x{}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D4)
Area (with t=edge length)	t2
Internal angle (degrees)	90°

In Euclidean geometry, a square is a regular polygon with four equal sides. For edge in Graph theory, see Edge (graph theory In Geometry, an edge is a one-dimensional Line segment joining In Geometry, a vertex (plural "vertices" is a special kind of point. In Mathematics, the Schläfli symbol is a notation of the form {pqr The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. Geometry, an interior angle (or internal angle) is an Angle formed by two sides of a Simple polygon that share an endpoint namely the angle This article describes the unit of angle For other meanings see Degree. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. General properties These properties apply to both convex and star regular polygons A square with vertices ABCD would be denoted ABCD.

Classification

A square (regular quadrilateral) is a special case of a rectangle as it has four right angles and equal parallel sides. In Geometry, a quadrilateral is a Polygon with four sides or edges and four vertices or corners. In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as Likewise it is also a special case of a rhombus, kite, parallelogram, and trapezoid. In Geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos “rhombus spinning top” (plural rhombi or rhombuses In Geometry a kite, or deltoid, is a Quadrilateral with two disjoint pairs of Congruent Adjacent sides in contrast In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides A trapezoid (in North America or a trapezium (in Britain and elsewhere is a Quadrilateral (a closed plane shape with four linear sides that has at least one

Mensuration formulae

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. The perimeter is the distance around a given two-dimensional object

And the area is

A = t². Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve.

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.

Properties

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

The diagonals of a square are equal. A diagonal can refer to a line joining two nonconsecutive vertices of a Polygon or Polyhedron, or in contexts any upward or downward sloping line Conversely, if the diagonals of a rhombus are equal, then that rhombus must be a square. A diagonal can refer to a line joining two nonconsecutive vertices of a Polygon or Polyhedron, or in contexts any upward or downward sloping line In Geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos “rhombus spinning top” (plural rhombi or rhombuses The diagonals of a square are (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. The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2}   or   √2 In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction

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 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). In Geometry, the Square tiling is a regular tiling of the Euclidean plane Plane tilings by Regular polygons have been widely used since antiquity Properties The area of an equilateral triangle with sides of length a\\! Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees
• The square is in two families of polytopes in two dimensions: hypercube and the cross polytope. In Geometry, a hypercube is an n -dimensional analogue of a square ( n = 2 and a Cube ( n = 3 In Geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular, convex Polytope that exists in any number of dimensions The Schläfli symbol for the square is {4}. In Mathematics, the Schläfli symbol is a notation of the form {pqr
• 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°. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. Its symmetry group is the dihedral group D₄. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections

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. Spherical geometry is the Geometry of the two- Dimensional surface of a Sphere. A great circle is a Circle on the surface of a Sphere that has the same circumference as the sphere dividing the sphere into two equal Hemispheres. 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. In 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. Geometry, an interior angle (or internal angle) is an Angle formed by two sides of a Simple polygon that share an endpoint namely the angle This is called a spherical cube. The Schläfli symbol is {4,3}. In Mathematics, the Schläfli symbol is a notation of the form {pqr

Squares can tile the Euclidean plane with 4 around each vertex, with each square having an internal angle of 90 degrees. In Geometry, the Square tiling is a regular tiling of the Euclidean plane Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. The Schläfli symbol is {4,4}. In Mathematics, the Schläfli symbol is a notation of the form {pqr

Squares can tile the hyperbolic plane with 5 around each vertex, with each square having 72 degree internal angles. In Geometry, the order-5 square tiling is a regular tiling of the Hyperbolic plane. In The Schläfli symbol is {4,5}. In Mathematics, the Schläfli symbol is a notation of the form {pqr

See also

• Cube
• Pythagorean theorem
• Square lattice
• Unit square

External links

• Square Calculation
• Animated course (Construction, Circumference, Area)
• Eric W. Weisstein, Square at MathWorld. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry In Mathematics, the square lattice is one of the five two-dimensional lattice types The unit square is a square with all of the side lengths equalling 1 Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• Definition and properties of a square With interactive applet
• Animated applet illustrating the area of a square