Archimedes' Limits to Value of Pi/Archimedes' Iterative Proof/Historical Note

Historical Note on Archimedes' Limits to Value of Pi: Archimedes' Iterative Proof

This proof is presented in the same way that Archimedes presented it.

He started the upper bound analysis by finding the perimeter of a circumscribed regular hexagon.

Then, using symmetry, one side of this hexagon is equated to twice the tangent of a right triangle with central angle $30 \degrees$ in a unit circle.

The lower bound analysis, on the other hand, starts by finding the perimeter of an inscribed regular hexagon as the sine of a right triangle with inscribed angle of $60 \degrees$ in a circle with unit diameter.

Using this technique it is seen that:

$(1): \quad$ the analyses both start with a regular hexagon, enabling the same sequence of polygons to be used

$(2): \quad$ the estimates of $\sqrt 3$ are initial inputs to both parts of the algorithm.

This serves to simplify the development.