Regions in a plane have a number of
interesting properties that one-dimensional
figures don't. Two such properties, and very important ones at that, are
perimeter and area. The perimeter of a region in a plane
is the length of the curve, or curves, that bound
the region. Finding the perimeter of a
polygon is easy; one needs only to sum the lengths
of the sides of the polygon. With
other curves, like circles for example, the length
of the boundary curve is more difficult
to calculate. There are formulas, though, that make the calculation of the
length of such curves easier. We'll take a look at these formulas.

The area of a region in a plane is the rough equivalent of the length of a
one-dimensional object: it is the most important and applicable property of
figures in that dimension, and is the main trait through which regions in a
plane can be compared. A region in a plane is formally defined as a simple
closed surface united with its interior
points. All points in a region are
coplanar, of course. The area of a region in
a plane is the number of square units the region covers. A square unit is a
unit of length and width. Each unit is an actual
square whose sides are of length one. With regions
bound by simpler curves, formulas exist for calculating area. Such regions
include polygons, whose sides are line
segments. Various polygons, including
regular polygons,
quadrilaterals, and most importantly,
triangles, have simple formulas for the calculation
of their area. However, because some regions are bound by simple closed curves
that aren't straight edges, the square units don't always nicely fit, making
calculation of the areas of those regions difficult. With complex
curves, calculus is needed to calculate area.

In the following lessons, we'll look at the more simple regions and study the
formulas that allow the calculation of those regions' areas. Eventually, with
knowledge of perimeter and area, we will build a repertoire of different ways to
compare geometric figures.