Name _________________________________
From Euclid to Spigotry
Assignment 9 (outline 9)
Last modified: September 14, 2004
You can do everything on this piece of paper and hand it in.  Feel free to use a calculator or Mathematica to obtain or check your answers.
1. Find exact expressions, in terms of square roots, for the perimeter p3 of an equilateral triangle inscribed in a circle of diameter 1 and the perimeter P3 of an equilateral triangle circumscribed about such a circle.  Then use Archimedes' formulas
=  and  =  to compute p6 and P6
The diagram below should be helpful.

2. For a circle of diameter 1, the perimeter of an inscribed polygon of n sides is pn = n sin(p/n) while the perimeter of a circumscribed polygon of n sides is Pn = n tan(p/n).
Show that tan (x/2) = and use this result to get Archimedes' formula for P2n in terms of Pn and pn.