SQUARE (GEOMETRY)

{| border="1" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Square
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|align=center colspan=2|
A square
''The sides of a square and its diagonals meet at right angles.''
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|bgcolor=#e7dcc3|Edges and vertices||4
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|bgcolor=#e7dcc3|Schläfli symbols||{4}
{}x{}
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|bgcolor=#e7dcc3|Coxeter–Dynkin diagrams||

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|bgcolor=#e7dcc3|Symmetry group||Dihedral (D₄)
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|bgcolor=#e7dcc3|Area
(with ''t''=edge length)||t²
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|bgcolor=#e7dcc3|Internal angle
(degrees)||90°
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In plane (Euclidean) geometry, a 'square' is a regular polygon with four sides.

Contents

Classification

Mensuration formulae

Standard coordinates

Properties

Other facts

Non-Euclidean geometry

See also

External links

Classification

A 'square' (regular quadrilateral) is a special case of a rectangle as it has four right angles and parallel sides. Likewise it is also a special case of a rhombus, kite, parallelogram, and trapezoid.

Mensuration formulae

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 (''x''₀, ''x''₁) 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. Conversely, if the diagonals of a rhombus are equal, then that rhombus must be a square. The diagonals of a square are $sqrt\{2\}$ (about 1.41) times the length of a side of the square. This value, known as Pythagoras’ constant, was the first number proven to be irrational.
If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths) then it is a square.

Other facts

★ If a circle is circumscribed around a square, the area of the circle is $pi/2$ (about 1.57) times the area of the square.

★ If a circle is inscribed in the square, the area of the circle is $pi/4$ (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 tiling 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 through 90°, 180° and 270°. Its symmetry group is the dihedral group $D\_4$.

★ If the area of a given square with side length S is multiplied by the area of a "unit triangle" (an equilateral triangle with side length of 1 unit), which is $frac\{sqrt\{3\}\}\{4\}$ units squared, the new area is that of the equilateral triangle with side length S.

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:'
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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}.
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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}.
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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}.
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See also

★ Pythagorean theorem

★ Square lattice

★ Square tiling

★ Unit square

External links

★ Animated course (Construction, Circumference, Area)

★

★ Definiton and properties of a square With interactive applet

★ Animated applet illustrating the area of a square