Estimating Circumference of a Circle
The applet below illustrates the manner in which the perimeters of inscribed regular polygons give an estimate to the circumference of a circle.
What if applet does not run? |
If C is the circumference of the circle, p_{n} an inscribed regular n-gon with perimeter P_{n}, is that true that
lim_{n→∞}P_{n} = C. |
As the number of sides n of the inscribed polygons grows without bound (i.e., "approaches ∞"), the perimeter (as a function of n) approaches a certain fixed value, which is naturally thought of as the perimeter (circumference) of the circle.
Why is that so? The fact has been established yet by Archimedes more than two thousand years ago. The perimeters of the inscribed n-gons form an increasing sequence. Archimedes also drew the circumscribed n-gons and showed that the perimeters of those form a decreasing sequence. Importantly, he also proved that the difference of the perimeters of the circumscribed and inscribed n-gons becomes increasingly smaller as n grows. (He gave estimates for n = 3, 6, 12, 24, 48, 96.) Archimedes did not possess a formal definition of limit, nor even a notion of real numbers. However, his approach can be easily formalized in a modern Calculus framework.
References
- S. Stein, Archimedes: What Did He Do Besides Cry Eurika?, MAA, 1999
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