linguistic:paraphrasing
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,

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 angles 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 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).