What is the area of a circle?

This question has already occupied the Egyptians around 2000 BC. This exhibit should help to find an answer to this question. By “skillfully” flipping sectors of a given circle, one can see that its area is exactly the same as that of a parallelogram-like figure (cf. Figure 3). In this way, one obtains a relationship between the circumference and the area of a circle.


And now … the mathematics of it:

First, the given circle is divided into n congruent circle sectors. Here n should be even (for odd n an analogous procedure, but with a “trapezoid” instead of a “parallelogram”, would lead to the same result). In our example n=12 was chosen.

Figure 1: The circle divided into twelve sectors

Jeder dieser Sektoren hat folgende Gestalt:

Figure 2: A circle sector

If you take a look at the exhibit, you will quickly realize that you can’t get anywhere with simple “trial and error” or even “force” in the task of determining the area of the circle, at least approximately. But there is a good approximate solution. This looks like this:

Figure 3: The solution offered by the exhibit

One now looks for correlations between the underlying circle and the associated “parallelogram” (Figure 3). It is valid:

  • The circle and the “parallelogram” have the same area.
  • The “parallelogram” becomes more and more like a rectangle with increasing n, whose height is the radius r of the given circle and whose width is equal to half the circumference p/2 of the same.

Why. Here are the reason(s):

  • Since the circle and the “parallelogram” are composed of the same parts, both must have the same area. Mathematically expressed: Equality of decomposition implies equality of content.
  • The sectors of the circle of the same size are getting closer and closer to a distance of length r as n grows. In the “parallelogram” they stand “upright”. Therefore, the height of the parallelogram approaches more and more the radius r (see figure 3).
  • If the sectors lie in the circle (cf. Figure 1), the circumference p of the circle corresponds to the product n\cdot b of the number n of sectors (in our example n=12) with the arc length b of a sector (cf. Figure 2). In the “parallelogram” (Figure 3), exactly half of the sectors are arranged with their tips pointing upwards and the other half in the opposite direction. This arrangement results in the length of the base side of the “parallelogram” approaching more and more the value \frac{n}{2}\cdot b=p/2.

If we summarize these considerations, we get for the area A of the parallelogram (which is equal to the area of the circle):

    \[A=a\cdot h=\frac{p}{2}\cdot r,\]

because — as justified above — for n\to\infty it approaches more and more a rectangle of height h=r and base side of length a=p/2. From this follows the formula for the circular area:

    \[A=\frac{p r}{2}.\]

This result can now be combined with the one from the exhibit [“What is Pi?”]. Namely, there you learn that

    \[\pi=\frac{p}{d}=\frac{p}{2r}.\]

Here d=2r is the diameter of the given circle. In total, the area of the circle of radius r is

    \[A=\frac{p r}{2}=\frac{p}{2r}\cdot r^2=\pi r^2\]

,

which corresponds to the known formula for the area of a circle.


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