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Square (geometry)
 
 
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In Euclidean geometry
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
, a square is a regular polygon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
 with four equal sides and four equal angles (90 degree
Degree (angle)

A degree , usually denoted by ? , is a measurement of plane angle, representing 1/360 of a Turn ; one degree is equivalent to p/180 radians....
 angles, or right angles). A square with vertices ABCD would be denoted .

All squares are parallelogram
Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
s.

dimensional object made up with four points, and four equal line-segments.

perimeter
Perimeter

A perimeter is a path that bounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length....
 of a square whose sides have length t is And the area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
 is

In classical times, the second power was described in terms of the area of a square, as in the above formula.






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In Euclidean geometry
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
, a square is a regular polygon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
 with four equal sides and four equal angles (90 degree
Degree (angle)

A degree , usually denoted by ? , is a measurement of plane angle, representing 1/360 of a Turn ; one degree is equivalent to p/180 radians....
 angles, or right angles). A square with vertices ABCD would be denoted .

All squares are parallelogram
Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
s.

Classification

Two-dimensional object made up with four points, and four equal line-segments.

Mensuration formula

The perimeter
Perimeter

A perimeter is a path that bounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length....
 of a square whose sides have length t is And the area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
 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.

Properties


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 diagonal
Diagonal

A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
s of a square are equal. Conversely, if the diagonal
Diagonal

A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
s of a rhombus
Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
 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
Square root of 2

The square root of 2, also known as Pythagoras' constant,is the positive real number that, when multiplied by itself, gives the number 2 ....
, was the first number proven to be irrational
Irrational number

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 m/n, where m and n are integers, with n non-zero....
.

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)

Other facts

  • If a circle is circumscribed around a square, the area of the circle is (about 1.57) times the area of the square.
  • If a circle is inscribed in the square, the area of the circle is (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
    Square tiling

    In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....
     is one of three regular tilings
    Tiling by regular polygons

    Plane Tessellation by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler in Harmonices Mundi....
     of the plane (the others are the equilateral triangle
    Equilateral triangle

    In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
     and the regular hexagon
    Hexagon

    In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
    ).
  • The square is in two families of polytopes in two dimensions: hypercube
    Hypercube

    In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
     and the cross polytope. The Schläfli symbol
    Schläfli symbol

    In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
     for the square is .
  • The square is a highly symmetric object (in Goldman geometry). There are four lines of reflectional symmetry
    Reflection symmetry

    The triangles with this symmetry are isosceles. The quadrilaterals with this symmetry are the kite s and the isosceles trapezoids.For each line or plane of reflection, the symmetry group is isomorphic with Cs , one of the three types of order two , hence algebraically C2....
     and it has rotational symmetry
    Rotational symmetry

    File:The armoured triskelion on the flag of the Isle of Man.svgGenerally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation....
     through 90°, 180° and 270°. Its symmetry group
    Symmetry group

    The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
     is the dihedral group
    Dihedral group

    In mathematics, a dihedral group is the group of symmetry of a regular polygon, including both rotational symmetry and reflection symmetry. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry....
     .


Non-Euclidean geometry

In non-euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles.

In spherical geometry
Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. Two practical applications of the principles of spherical geometry are navigation and astronomy....
, a square is a polygon whose edges are great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
 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
Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P th...
, 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 angle
Internal angle

In geometry, an interior angle is an angle formed by two sides of a simple polygon that share an endpoint, namely, the angle on the inner side of the polygon....
s. This is called a spherical cube. The Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 is .
Squares can tile
Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....
 the Euclidean plane with 4 around each vertex, with each square having an internal angle of 90 degrees. The Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 is .
Squares can tile
Order-5 square tiling

In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....
 the hyperbolic plane
Hyperbolic plane

In mathematics, the term hyperbolic plane may refer to:* A two-dimensional quadratic space with a non-singular isotropic quadratic form* A plane in hyperbolic geometry...
 with 5 around each vertex, with each square having 72 degree internal angles. The Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 is .


See also

  • Cube
    Cube

    A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
  • Pythagorean theorem
    Pythagorean theorem

    In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
  • Square lattice
    Square lattice

    In mathematics, the square lattice is one of the five two-dimensional Lattice types. It is the two-dimensional version of the integer lattice....
  • Unit square
    Unit square

    The unit square is a square with all of the side lengths equalling 1....


External links

  • With interactive applet