Square (geometry)

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

Classification

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

Mensuration formula

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

*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] ).
*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 through 90°, 180° and 270°. Its symmetry group is the dihedral 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 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:

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)]
*
* [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]