Editing Square (section)

{{Short description|Shape with four equal sides and angles}}
{{CS1 config|mode=cs1}}
{{About|the shape}}{{Infobox polygon
| name       = Square
| image      = Regular polygon 4 annotated.svg
| caption    =
| type       = {{plainlist|1=
*[[quadrilateral]]
*[[regular polygon]]
*[[hypercube]]
*[[cross-polytope]]
}}
| euler      =
| edges      = 4
| schläfli   = 
| wythoff    = 
| coxeter    = 
| symmetry   = [[Dihedral group of order 8|order-8 dihedral]]
| area       = side<sup>2</sup>
| angle      = {{pi}}/2 (90°)
| perimeter  = 4 · side
| dual       = 
| properties =
}}
In [[geometry]], a '''square''' is a [[regular polygon|regular]] [[quadrilateral]]. It has four straight sides of equal length and four equal [[angle]]s. Squares are special cases of [[rectangle]]s, which have four equal angles, and of [[rhombus]]es, which have four equal sides. As with all rectangles, a square's angles are [[right angle]]s (90 [[degree (angle)|degree]]s, or [[Pi|{{pi}}]]/2 [[radian]]s), making adjacent sides [[perpendicular]]. The [[area]] of a square is the side length multiplied by itself, and so in [[algebra]], multiplying a number by itself is called [[square (algebra)|squaring]].

Equal squares can tile the plane edge-to-edge in the [[square tiling]]. Square tilings are ubiquitous in [[tile]]d floors and walls, [[graph paper]], image [[pixel]]s, and [[game board]]s. Square shapes are also often seen in building [[floor plan]]s, [[origami paper]], food servings, in [[graphic design]] and [[heraldry]], and in instant photos and fine art.

The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for [[squaring the circle]] by [[compass and straightedge]], now known to be impossible. Squares can be inscribed in any smooth or convex curve such as a circle or triangle, but it remains unsolved [[inscribed square problem|whether a square can be inscribed in every simple closed curve]]. Several problems of [[squaring the square]] involve subdividing squares into unequal squares. Mathematicians have also studied packing squares as tightly as possible into other shapes.

Squares can be constructed by [[straightedge and compass]], through their [[Cartesian coordinates]], or by repeated multiplication by <math>i</math> in the [[complex plane]]. They form the [[Ball (mathematics)|metric balls]] for [[taxicab geometry]] and [[Chebyshev distance]], two forms of non-Euclidean geometry. Although [[spherical geometry]] and [[hyperbolic geometry]] both lack polygons with four equal sides and right angles, they have square-like regular polygons with four sides and other angles, or with right angles and different numbers of sides.