Archimedes' Method, Part III

Recall our construction:

We have seen that = sin and = tan. Since we have used  -gons, it follows that = sin() and = tan().

In order to relate and to and , we use two trig identities:


In terms of the perimeters of our polygons, this means:



Notice that is the harmonic mean of and , while is the geometric mean of and . (Sound familiar?)

Finally, let's look at a simple example.

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Neal Carothers -