Square  

A square The sides of a square and its diagonals meet at right angles. 

Edges and vertices  4 
Schläfli symbols  {4} {}x{} 
Coxeter–Dynkin diagrams  Image:CDW_ring.pngImage:CDW_ring.png 
Symmetry group  Dihedral (D_{4}) 
Area (with t=edge length) 
t^{2} 
Internal angle (degrees) 
90° 
In plane (Euclidean) geometry, a square is a regular polygon with four sides. It may also be thought of as 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.
Contents 
The perimeter of a square whose sides have length t is
And the area is
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.
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.
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 <math>\sqrt{2}</math> (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.
In noneuclidean geometry, squares are more generally polygons with four 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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