# Areas

#### Introduction Area is a property of alltwo-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, althougharea 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 assurface area.

A region in a plane isdefined as any simple closed curve united with its interior. Such a curve canbe convex or concave; either way, it has area. The unit of measurement of areais the square unit, which, specifically, is a square whose sides are one unitlong. Square units are a generic term; it can be measured according todifferent measures of length. For example, a piece of paper is measured insquare 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 somethinglike this:

A region, bound by anysimple 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 agrid of square units is placed over a region whose sides aren't straight, areabecomes easier to visualize. The grid makes it possible to count the squareunits and estimate the fractions of square units in the region and approximateits area. Here is how the technique is employed:

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

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

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