Challenge Level

By inscribing a circle in a square and then a square in a circle
find an approximation to pi.

By using a hexagon, can you improve on the approximation? How much
better an approximation is it?

Archimedes used this idea first with a hexagon, then a dodecagon (12 sides) and so on up to a 96 sided polygon to calculate pi and was able to establish that $$3\frac{10}{71} < \pi < 3 \frac{1}{7}$$

What are the strengths and limitations of this method?