The Area of a Circle - Proof and Understanding

Discussing the area formula of a circle must involve three separate, though interrelated discussions. First a working definition of "area" is required. The number pi is necessary in most calculations involving circles, and certainly appears in the area formula. Then, once the background has been covered, the actual area formula can be discussed with confidence.

The "American Heritage Dictionary" provides us with a working definition of area. It says that area is "the extent of a planar region or of the surface of a solid measured in square units.(1)" For our purposes, the area includes the boundary of a figure and all points within. In the case of a circle this means that area will encompass all points that are located at a distance from the center of the circle that is less than or equal to the radius of the circle. One can envision that the area of any figure could be determined by filling it full of squares of known dimensions. (This is not unlike the concept of pixilation used to display images on computer screens.) For a square, area is given as the square of the length of one side. Since measurements of length have units, the area then has units of length squared. Any equation that is used to find area (filling objects with infinitely small squares and counting them to determine area is impractical) must also have dimensions of length squared. Such is the case for rectangles (length X width), triangles (1/2 base X height), and circles as well, with the traditional formula of pi*(r^2).

The number pi represents a fixed ratio in circles. Specifically, it is the ratio of the circumference of a circle to the diameter of the same circle. Historically, approximations have been used for pi, since it is an irrational and transcendental number. Various infinite series can be used to calculate a portion of pi. In modern times, computers have made it possible to compute its value beyond any useful need for precision. (Yasumasa Kanada computed it to over one trillion places in 2002.(2)) Numerous such computer algorithms exist, and are easily found on the internet. pi is very relevant to a discussion of the circle area formula, not only because it appears in the formula, but because using that formula was how Archimedes first approximated its value, using the areas of many-sided polygons to set upper and lower limits for pi.(3,4) This will be discussed further as one approach to proving the

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