Editing Square (section)

===Measurement===
[[File:YBC-7289-OBV-labeled.jpg|[[YBC 7289]], a [[Babylonian mathematics|Babylonian]] calculation of a square's diagonal from between 1800 and 1600 BCE|thumb]]
[[Image:Five Squared.svg|150px|right|thumb|The area of a square is the product of the lengths of its sides.]]

A square whose four sides have length <math>\ell </math> has [[perimeter]]{{sfnp|Rich|1963|p=[https://archive.org/details/in.ernet.dli.2015.131938/page/n147 131]}} <math>P=4\ell</math> and [[diagonal]] length <math>d=\sqrt2\ell</math>.{{sfnp|Rich|1963|p=[https://archive.org/details/in.ernet.dli.2015.131938/page/n135 120]}} The [[square root of 2]], appearing in this formula, is [[irrational number|irrational]], meaning that it is not the ratio of any two [[integer]]s. It is approximately equal to 1.414,<ref>{{cite book |last1=Conway |first1=J. H. |author-link1=John Horton Conway |last2=Guy |first2=R. K. |author-link2=Richard K. Guy |title=The Book of Numbers |title-link=The Book of Numbers (math book) |year=1996 |publisher=Springer-Verlag |location=New York|pages=181–183}}</ref> and its approximate value was already known in [[Babylonian mathematics]].<ref>{{cite journal
 | last1 = Fowler | first1 = David | author1-link = David Fowler (mathematician)
 | last2 = Robson | first2 = Eleanor | author2-link = Eleanor Robson
 | doi = 10.1006/hmat.1998.2209
 | issue = 4
 | journal = [[Historia Mathematica]]
 | mr = 1662496
 | pages = 366–378
 | title = Square root approximations in old Babylonian mathematics: YBC 7289 in context
 | volume = 25
 | year = 1998| doi-access = free
 }}</ref> A square's [[area]] is{{sfnp|Rich|1963|p=[https://archive.org/details/in.ernet.dli.2015.131938/page/n135 120]}}
<math display="block">A=\ell^2=\tfrac12 d^2.</math>
This formula for the area of a square as the second power of its side length led to the use of the term ''[[Square (algebra)|squaring]]'' to mean raising any number to the second power.<ref>{{cite book|first=James|last=Thomson|author-link=James Thomson (mathematician)|title=An Elementary Treatise on Algebra: Theoretical and Practical|year=1845|location=London|publisher=Longman, Brown, Green, and Longmans|page=4|url=https://archive.org/details/anelementarytre01thomgoog/page/n15}}</ref> Reversing this relation, the side length of a square of a given area is the [[square root]] of the area. Squaring an integer, or taking the area of a square with integer sides, results in a [[square number]]; these are [[figurate number]]s representing the numbers of points that can be arranged into a square grid.{{sfnp|Conway|Guy|1996|pp=30–33,38–40}}

Since four squared equals sixteen, a four by four square has an area equal to its perimeter. That is, it is an [[equable shape]]. The only other equable integer rectangle is a three by six rectangle.<ref>{{cite book |last1=Konhauser |first1=Joseph D. E. | author1-link=Joseph Konhauser |last2=Velleman |first2=Dan |last3=Wagon |first3=Stan |authorlink3=Stan Wagon |date=1997 |title=Which Way Did the Bicycle Go?: And Other Intriguing Mathematical Mysteries |contribution=95. When does the perimeter equal the area? |volume=18 |series=Dolciani Mathematical Expositions |publisher=Cambridge University Press |isbn=9780883853252 |page=29 |url=https://books.google.com/books?id=ElSi5V5uS2MC&pg=PA29}}</ref>

Because it is a [[regular polygon]], a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter.<ref>Page 147 of {{cite book
 | last = Chakerian | first = G. D.
 | editor-last = Honsberger | editor-first = Ross | editor-link = Ross Honsberger
 | contribution = A distorted view of geometry
 | isbn = 0-88385-304-3
 | mr = 563059
 | pages = 130–150
 | publisher = Mathematical Association of America | location = Washington, DC
 | series = The Dolciani Mathematical Expositions
 | title = Mathematical Plums
 | volume = 4
 | year = 1979}}</ref>  Indeed, if ''A'' and ''P'' are the area and perimeter enclosed by a quadrilateral, then the following [[isoperimetric inequality]] holds:
<math display="block">16A\le P^2</math>
with equality if and only if the quadrilateral is a square.<ref>{{cite journal
 | last = Fink | first = A. M.
 | date = November 2014
 | doi = 10.1017/S0025557200008275
 | issue = 543
 | journal = [[The Mathematical Gazette]]
 | jstor = 24496543
 | page = 504
 | title = 98.30 The isoperimetric inequality for quadrilaterals
 | volume = 98}}</ref>{{sfnp|Alsina|Nelsen|2020|loc=Theorem 9.2.2|page=187}}