Limits of sequences

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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?

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?