Editing Area (section)

===Circle area===
{{main|Area of a circle#History}}

In the 5th century BCE, [[Hippocrates of Chios]] was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his [[Quadrature (geometry)|quadrature]] of the [[lune of Hippocrates]],<ref name="heath">{{citation|first=Thomas L.|last=Heath|author-link=Thomas Little Heath|title=A Manual of Greek Mathematics|publisher=Courier Dover Publications|year=2003|isbn=978-0-486-43231-1|pages=121–132|url=https://books.google.com/books?id=_HZNr_mGFzQC&pg=PA121|url-status=live|archive-url=https://web.archive.org/web/20160501215852/https://books.google.com/books?id=_HZNr_mGFzQC&pg=PA121|archive-date=2016-05-01}}</ref> but did not identify the [[constant of proportionality]]. [[Eudoxus of Cnidus]], also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared.<ref>{{cite book|url=https://archive.org/details/singlevariableca00stew/page/3|title=Single variable calculus early transcendentals.|last=Stewart|first=James|publisher=Brook/Cole|year=2003|isbn=978-0-534-39330-4|edition=5th.|location=Toronto ON|page=[https://archive.org/details/singlevariableca00stew/page/3 3]|quote=However, by indirect reasoning, Eudoxus (fifth century B.C.) used exhaustion to prove the familiar formula for the area of a circle: <math>A= \pi r^2.</math>}} <!--This quote may be an overstatement. I have not been able to confirm that he discovered the actual formula, but perhaps only the proportionality between A and r-squared.--></ref>

Subsequently, Book I of [[Euclid's Elements|Euclid's ''Elements'']] dealt with equality of areas between two-dimensional figures. The mathematician [[Archimedes]] used the tools of [[Euclidean geometry]] to show that the area inside a circle is equal to that of a [[right triangle]] whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book ''[[Measurement of a Circle]]''. (The circumference is 2{{pi}}''r'', and the area of a triangle is half the base times the height, yielding the area {{pi}}''r''<sup>2</sup> for the disk.) Archimedes approximated the value of {{pi}} (and hence the area of a unit-radius circle) with [[Area of a disk#Archimedes' doubling method|his doubling method]], in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular [[hexagon]], then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with [[circumscribed polygon]]s).