CTK Insights

03 Feb

What happens to the area when the radius of circle is doubled?

Assume you work with the kids who do not yet know the famous formula \pi R^2 for the area of a circle of radius R. How would you explain to the kids that the area of the circle quadruples when the radius doubles? This is the question raised by Linda Fahlberg-Stojanovska at the mathfuture google group. For a square, the quadrupling of the area is easy to visualize, but for a circle? To answer the question Linda created a beautiful GeoGebra applet, based on the ideas of Leonardo da Vinci and Sato Moshun.

Other approaches have been also brought up. One illustrates the distinction between the indivisibles (of Cavalieri and others) and infinitesimals of Newton, Leibniz, and Abraham Robinson.

Infinitesimals are (infinitely) small objects of the dimension of the space in which they are being used. In the da Vinci's method, the circle is 2-dimensional as are the sectoral slices even though each of the pieces - in the limit - has zero area.

Indivisibles are pieces of a whole that are 1 dimension less than the whole. For example, every triangle can be thought of as the union of linear segments parallel to the base. Each of the segments can be assigned length, and, according to Cavalier's principle, two triangles that stack the same line segments have the same area.

An application of indivisibles to the question at hand is based on the idea of Rabbi Abraham bar Hiyya Hanasi, who saw the circle as consisting of layers of circumferences, onion-like. When layers are peeled, straightened, and stacked, they form a triangle whose area is the same as that of the circle.

When the radius is doubled and the circle is peeled into a triangle, the originally small circle maps onto the upper triangular portion of the big one. The big triangle consists of four such pieces.

It is also possible to indirectly use the argument by similarity. A circle of radius R is inscribed into a square of side 2R. Let C be the area of a circle, S that of the square. I believe it is intuitively clear that the ratio \displaystyle\frac{S-C}{C} is independent of R. Defining it as, say, \displaystyle k=\frac{S-C}{C} gives \displaystyle C=\frac{S}{k+1}, with the conclusion that, if S changes by a factor of 4, so does C.

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