Editing Area (section)

===Relation of area to perimeter===

The [[isoperimetric inequality]] states that, for a closed curve of length ''L'' (so the region it encloses has [[perimeter]] ''L'') and for area ''A'' of the region that it encloses,

:<math>4\pi A \le L^2,</math>

and  equality holds if and only if the curve is a [[circle]]. Thus a circle has the largest area of any closed figure with a given perimeter.

At the other extreme, a figure with given perimeter ''L'' could have an arbitrarily small area, as illustrated by a [[rhombus]] that is "tipped over" arbitrarily far so that two of its [[angle]]s are arbitrarily close to 0° and the other two are arbitrarily close to 180°.

For a circle, the ratio of the area to the [[circumference]] (the term for the perimeter of a circle) equals half the [[radius]] ''r''. This can be seen from the area formula ''πr''<sup>2</sup> and the circumference formula 2''πr''.

The area of a [[regular polygon]] is half its perimeter times the [[apothem]] (where the apothem is the distance from the center to the nearest point on any side).