- Square (geometry)
In

Euclidean Geometry geometry , a**square**is aregular polygon with four equal sides. InEuclidean geometry , it has four 90 degree angles. A square with vertices ABCD would be denoted squarenotation|ABCD.**Classification**A

**square**(regularquadrilateral ) is a special case of arectangle as it has four right angles and equal parallel sides. Likewise it is also a special case of arhombus , kite,parallelogram , andtrapezoid .**Mensuration formula**The

perimeter of a square whose sides have length "t" is :$P=4t.$And thearea 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

diagonal s of a square are equal. Conversely, if thediagonal s of arhombus 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***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 $pi/2$ (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 ( [*http://www2.mat.dtu.dk/people/V.L.Hansen/square.html*] ).

*Asquare tiling is one of three regular tilings of the plane (the others are theequilateral triangle and the regular hexagon).

*The square is in two families of polytopes in two dimensions:hypercube and thecross polytope . TheSchläfli 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 hasrotational symmetry through 90°, 180° and 270°. Itssymmetry group is thedihedral group $D\_4$.**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 aregreat 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:****ee also***

Cube

*Pythagorean theorem

*Square lattice

*Unit square **External links*** [

*http://easycalculation.com/area/square.php Square Calculation*]

* [*http://www.elsy.at/kurse/index.php?kurs=Rectangle+and+Square&status=public Animated course (Construction, Circumference, Area)*]

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* [*http://www.mathopenref.com/square.html Definition and properties of a square*] With interactive applet

* [*http://www.mathopenref.com/squarearea.html Animated applet illustrating the area of a square*]

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