Archimedes' Method, Part II

Recall that we want to estimate the circumference of a circle of diameter 1 (which we know to be ). For each = 2, 3, 4, ..., we inscribe and circumscribe regular polygons having sides.

Recall, too, that the perimeters and satisfy .

In order to generate an iterative formula for the perimeters, we use a bit of geometry:

If we denote the central angle in our  -gon by 2, then is the side opposite the angle in a right triangle with hypoteneuse 1. Hence, = sin.

Next we rotate our picture and concentrate on .

This time our right triangle has as the side opposite the angle , and as the adjacent side. Hence, = tan.

Archimedes' iterative formulas for and will now follow from two trig identities.

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Neal Carothers -