Area is a property of all
two-dimensional
figures. It measures the combined length and width of a region. In the
following lessons we'll explore the area of regions in a
plane, although area is also a property of
two-dimensional surfaces that don't lie in
a plane. In those cases, covered in Three Dimensional
Measurements, it is referred to as surface
area.

A region in a plane is defined as any simple closed
curve united with its interior. Such a curve can be
convex or concave;
either way, it has area. The unit of measurement of area is the square
unit, which, specifically, is a square whose
sides are one unit long. Square units is a generic
term; it can be measured according to different measures of length. For
example, a piece of paper is measured in square inches, whereas land is measured
in square miles. In this text, however, we'll just use the generic term square
units. A square unit looks something like this:

A region, bound by any simple closed curve, doesn't always break down into
squares of the same size; in fact, this kind of perfect break down happens very
rarely. There is a way, however, to make a decent approximation of the area of
such a region. When a grid of square units is placed over a region whose sides
aren't straight, area becomes easier to visualize. The grid makes it possible
to count the square units and estimate the fractions of square units in the
region and approximate its area. Here is how the technique is employed:

From an illustration like this one, it is relatively easy to approximate that
the area of the region is about 15 square units.

In geometry, we'll study cases in which a region does break down nicely
into squares. We'll also study cases in which a region breaks down into other
shapes, like triangles, whose areas can be
calculated using formulas. All of our study will hopefully make it possible to
make educated approximations of areas in real life.