Measuring Two-Dimensional Figures
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.