CTK Insights

The fact that π exists is due to the similarity of all circles. The ratio of the circumference to the diameter would not be constant otherwise.

The simple idea of similarity is commonly being reported as overly hard on children. I am confident that the problem must lie with the manner of presentation and not with the similarity per se. This is because children grow up in the world of models: toy cars, stuffed animals, house drawings, etc. To detect pattern - an activity in which the human brain has no peer - is to see similarities. This is to say that the notion of similarity should not be too difficult for the kids, i.e., if it is introduced properly. The difficulty usually lies in the algebraic expression of similarity not in the geometric concept.

I was very pleasantly surprised by a submission to the latest Math teachers at play carnival that featured a gentle introduction into the concept of similarity. Once explained and internalized, the concept was applied to the determination of π.

Here I would like to add my 2¢. It is a common practice to measure and tabulate the diameter and the circumference of several round objects and take the ratio. "Lo and behold" goes the argument, "the results are so close that not only may we estimate π but also convince ourselves that this fellow really exists."

Such experimentation goes a long way as a verification of an abstract concept of the similarity of all circles. However, we can easily do much better in estimating the value of π.

Let D and C be a measurement of the radius and of the circumference of a circular object with D₀ and C₀ being the real (but unknown) values. We assume that D is close to D₀ and C is close to C₀ and hope further that the ratio C/D is close to C₀D₀.

The accuracy of the measurement, i.e., the estimate for |C₀ - C| depends only on the measuring device. For example, if a ruler is used then the maximum error between the "real" and measured values may be assumed to be one half of the smallest division marked on the ruler. This is true for "big" and "small" lengths. If that max error is denoted Δ then |C₀ - C| < Δ, for any circumference C.

How do we measure circumference? Simple: take a thread and wrap it once around the object, unwrap it, keep it stretched and measure the length of the thread. Obviously, if we wrap the thread several times over then the measurements will have to be divided by the number of turns the thread was wrapped around the object. If n is the number of turn then the the measured length is close to nC. How close? The estimate is still within the same precision marks as it was for a single turn, i.e., Δ. For an estimate C₀ we then have |nC₀ - nC| < Δ. As a consequence |C₀ - C| < Δ/n.

Let's see how it works. For a round object I took a carton tube, a leftover from a roll of paper towels:

I wrapped a thread around the tube several times and made the marks:

The diameter was measured to be D = 13/8. (All measurements are in inches.) Then I tabulated the results for 1, 2, 3, and 4 wraps of the thread:

(The numbers in the bottom row have been rounded to 4 digits.)

I did not have a long enough ruler to measure 5 or more turns but the trend could be easily observed with just four measurements: the ratios go closer to π as the number of turns of the thread around the object grows.

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