Editing Circle (section)

===Area enclosed===
[[Image:Circle Area.svg|thumb|Area enclosed by a circle = {{pi}} × area of the shaded square]]
{{Main article|Area of a circle}}
As proved by [[Archimedes]], in his [[Measurement of a Circle]], the [[Area of a disk|area enclosed by a circle]] is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius,<ref>{{citation|first=Victor J.|last=Katz|title=A History of Mathematics / An Introduction|edition=2nd|year=1998|publisher=Addison Wesley Longman|isbn=978-0-321-01618-8|page=[https://archive.org/details/historyofmathema00katz/page/108 108]|url-access=registration|url=https://archive.org/details/historyofmathema00katz/page/108}}</ref> which comes to {{pi}} multiplied by the radius squared:
<math display="block">\mathrm{Area} = \pi r^2.</math>

Equivalently, denoting diameter by ''d'',
<math display="block">\mathrm{Area} = \frac{\pi d^2}{4} \approx 0.7854 d^2,</math>
that is, approximately 79% of the [[Circumscribe|circumscribing]] square (whose side is of length ''d'').

The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the [[isoperimetric inequality]].