In Measurement of the Circle, the great
Archimedes
(*c*. 287--212 BC) found an approximation for the
circumference of a circle of a given radius.

Since we know that the circumference and diameter of any circle are related by the formula , this means that if we start with a circle of diameter 1, then Archimedes' approximation for actually provides an approximation for .

Archimedes' idea was to approximate the circle using both inscribed and circumscribed (regular) polygons. Below are pictured inscribed and circumscribed octagons.

More generally, we would consider inscribed and circumscribed
-gons.
The inscribed -gon
has
sides, each of the same length ,
and the circumscribed -gon
has
sides, each of the same length .
(In truth, we should consider
-gons, where
*M* is a positive integer. But, for simplicity, forego this extra
generality.)

The perimeter of the inscribed -gon, which we denote by , and the perimeter of the circumscribed -gon, which we denote by , are approximations for and so, in this case, are also approximations for :

By means of geometric (and what we would now call trigonometric) arguments,
Archimedes was able to **derive iterative formulas** for
and
,
which are reminiscent of the
Babylonian algorithm for computing
square roots.

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Neal Carothers - carother@bgnet.bgsu.edu