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.

If C is the circumference of the circle, p_n an inscribed regular n-gon with perimeter P_n, is that true that

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

1. S. Stein, Archimedes: What Did He Do Besides Cry Eurika?, MAA, 1999

Copyright © 1996-2009 Alexander Bogomolny