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 circle area formula.
Archimedes is credited as the first person to make a significant attempt at finding pi and finding the area of a circle, and he did so by estimating the area of a circle using polygons of increasing number of sides. Any regular polygon of “n” sides inscribed on a circle can be divided into “n” isosceles triangles with the odd vertex at the center of the circle and the two equal vertices on the triangle. The base of each triangle cuts a small section of the circle out of the calculated area, with that portion diminishing as the number of sides increases. The base (b) and height (h) of the triangle can be found since the hypotenuse (r) is known and the central angle (O – formed by the radius and the altitude/height of the triangle) can be determined from the number of sides. The total area of all the triangles is found by multiplying the area of each triangle by the number of sides in the original polygon. In other words, the area of the circle is estimated using:
b = 2*r*sin O
h = r*cos O
O = 360 / (2*n)
n = number of sides
Area = n*(1/2)*b*h = (r^2)*sin O*cos O = (r^2)*sin (2*O)/2.
Allowing for a polygon with infinite sides – a circle – the height is the same as the radius and n*b is the circumference of the circle, or 2*pi*r, which simplifies to the familiar area formula Area = pi*(r^2). A similar process can be followed, using a polygon whose bases are each tangent to the circle at their midpoints. In this case, the height is the radius, and the resulting area is slightly larger than the circle’s, with the limiting case of the infinitely sided polygon again simplifying to pi*(r^2).(3,5)
There are some interesting approaches to the area formula that rely on a rearrangement of the circle into another geometric form. For example, Rabbi Abraham Bar Hiya demonstrated that the circle could be “unrolled” in a series of infinitesimal layers. Each layer would have length 2*pi*r, where r is the distance of that layer from the center of the circle. The resultant figure would be a triangle whose base was the circumference of the circle and whose height was the radius of the circle. This, of course, gives and area of 1/2*b*h = 1/2*2*pi*r*r = pi*(r^2).(6)
A second such tactic involves dividing the circle into wedges and placing them side by side, alternately pointing up and down. As the wedges get smaller (and more numerous), the resultant figure begins to resemble a rectangle. In the limiting case, the figure actually would be a rectangle. Since two of the sides are the sides of the wedges, they are the length of the radius. The other two sides are made up of the outer edges of the wedges, with a total length of 2*pi*r, so each is exactly pi*r long. Length times width, the area of a rectangle, gives the expected value of pi*(r^2).(7)
Such illustrations are comforting visualizations, but lack some of the rigor a purely mathematical proof would have. Textual limitations prevent the actual presentation here, but a rigorous proof can be seen at the following website:
http://www.artofproblemsolving.com/LaTeX/Examples/AreaOfACircle.pdf
which gives a proof stemming from the equation of a circle centered at the origin. The equation is integrated for the region under the curve in the first quadrant only. This gives an area of pi*(r^2)/4, as would be expected for a quarter circle.
This approach is much more elegant, all the more so because it does not rely upon a foreknowledge of the circumference formula for a circle. (Each of the other methods presented have, at least in the limiting cases.) While pi necessarily appears in this equation as well, its appearance is the result of radian-based angle measures, not circumference.
This examination of the area of a circle has covered all aspects of the topic, as pertaining to Euclidean geometry. (This includes the concept of area, pi, and both justifications and a rigorous proof of the circle area formula.) In alternate geometries, area may be calculated differently, and different formulas may apply. (Such is the case in spherical geometry, for instance.) This is beyond the scope of this article however, which was specifically concerned with A = pi*(r^2).
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Bibliography
1.
area. (n.d.). The American Heritage Dictionary of the English Language, Fourth Edition. Retrieved March 15, 2008, from Dictionary.com website: http://dictionary.reference.com/browse/area
2.
Vincent G. “What is the absolute value of pi at the current time?”
Retrieved March 15, 2008, from:
http://answers.yahoo.com/question/index?qid=20061014150814AAp0Qwc
3.
“How Archimedes Found the Area of a Circle”
Retrieved March 15, 2008, from:
http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/archimedes.html
4.
Dr. Wilkinson. “What is the Formula for pi?”
Retrieved March 15, 2008, from:
http://mathforum.org/library/drmath/view/52543.html
5.
“Area of a Disc”. Wikipedia
Retrieved March 15, 2008, from:
http://en.wikipedia.org/wiki/Area_of_a_disk
6.
Tsaban, Boaz and Garber, David. “The Proof of Rabbi Abraham Bar Hiya Hanasi” 2001
Retrieved March 15, 2008, from:
http://www.cs.biu.ac.il/~tsaban/Circles.html
7.
Willis, Bill. “A=pir2, Where Does it Come From?”. 1999
Retrieved March 15, 2008, from:
http://worsleyschool.net/science/files/circle/area.html
8.
nr1337. “Proof of the Area of a Circle Formula A = pir2”
Retrieved March 15, 2008, from:
http://www.artofproblemsolving.com/LaTeX/Examples/AreaOfACircle.pdf
