Editing Square (section)

==Definitions and characterizations==
[[File:The square among a family of rectangles or rhombuses.png|thumb|upright=1.25|Among rectangles (top row), the square is the shape with equal sides (blue, middle). Among rhombuses (bottom row), the square is the shape with right angles (blue, middle).]]
Squares can be defined or characterized in many equivalent ways. If a [[polygon]] in the [[Euclidean plane]] satisfies any one of the following criteria, it satisfies all of them:
* A square is a polygon with four equal sides and four [[right angle]]s; that is, it is a quadrilateral that is both a rhombus and a rectangle<ref name=zalgri/>
* A square is a [[rectangle]] with four equal sides.<ref name=zalgri>{{cite book|first1=Zalman|last1=Usiskin|author1-link=Zalman Usiskin|first2=Jennifer|last2=Griffin|title=The Classification of Quadrilaterals: A Study of Definition|publisher=Information Age Publishing|year=2008|page=59|isbn=978-1-59311-695-8}}</ref>
* A square is a [[rhombus]] with a right angle between a pair of adjacent sides.<ref name=zalgri/>
* A square is a rhombus with all angles equal.<ref name=zalgri/>
* A square is a [[parallelogram]] with one right angle and two adjacent equal sides.<ref name=zalgri/>
* A square is a  quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other. That is, it is a rhombus with equal diagonals.<ref>{{Cite web|url=http://jwilson.coe.uga.edu/MATH7200/ProblemSet1.3.html|title=Problem Set 1.3, problem 10|work=Math 5200/7200 Foundations of Geometry I|first=Jim|last=Wilson|publisher=University of Georgia|date=Summer 2010|access-date=2025-02-05}}</ref>
* A square is a  quadrilateral with successive sides <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> whose area is<ref>{{cite book
 | last1 = Alsina | first1 = Claudi
 | last2 = Nelsen | first2 = Roger B.
 | contribution = Theorem 9.2.1
 | isbn = 9781470453121
 | page = 186
 | publisher = American Mathematical Society
 | series = Dolciani Mathematical Expositions
 | title = A Cornucopia of Quadrilaterals
 | url = https://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA22
 | volume = 55
 | year = 2020}}</ref> <math display=block>A=\frac14(a^2+b^2+c^2+d^2).</math>

Squares are the only [[regular polygon]]s whose [[Internal and external angle|internal angle]], [[central angle#Central angle of a regular polygon|central angle]], and [[Internal and external angle|external angle]] are all equal (they are all right angles).<ref name=rich/>