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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:

Figure %: A square unit

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:

Figure %: A grid superimposed over a curve
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.

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