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In Plane Geometry , a square is a Polygon with four equal sides and equal angles.


CLASSIFICATION

Squares are a special case of Regular Quadrilateral , Rectangle , Rhombus , Kite , Parallelogram , and Isosceles Trapezoid/trapezium .


MENSURATION FORMULAE

The Perimeter of a square whose sides have length s is
:P=4s
And the Area is
:A=s^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

The angles of a square are necessarily Right Angles .

The Diagonal s 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 known to be Irrational .

If a figure is both a rectangle and a rhombus, then it is a square.


OTHER FACTS

  • If a circle is circumscribed around a square, the area of the circle is \pi/\sqrt{2} (about 2.22) 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 rectangle or rhombus with the same perimeter.

  • A square is one of three Regular Polygon s that can form a regular Tiling of the plane (the others are the Equilateral Triangle and the Regular Hexagon ). This is a consequence of the fact that the measure of the angles is a factor of 360°.

  • The square is both the Measure Polytope and the Cross Polytope in two dimensions. The Schläfli Symbol for the square is {4}.



NON-EUCLIDEAN GEOMETRY

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.


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