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In Plane (Euclidean) Geometry , a square is a Regular Polygon with four sides.


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 Diagonal s of a square are equal. Conversely, if the Diagonal s 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 ( {Link without Title} ).

  • 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:


SEE ALSO



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