Squaring the circle the circular arc b around O with radius , it results S'. Bisect the line segment in D and draw the semicircle b over D. Draw a straight line from O through C up to the semicircle b, it cuts b in E. The line segment is the mean proportional between and , also called geometric mean. Extend the line segment beyond O and transfer twice more, it results F and A, and thus the length of the line segment with the above described approximation value of formula_3, the half circumference of the circle. Bisect the line segment in G
Area of a circle good approximation to the circumference. To compute "u" and "U" for large "n", Archimedes derived the following doubling formulae: Starting from a hexagon, Archimedes doubled "n" four times to get a 96-gon, which gave him a good approximation to the circumference of the circle. In modern notation, we can reproduce his computation (and go further) as follows. For a unit circle, an inscribed hexagon has "u" = 6, and a circumscribed hexagon has "U" = 4. Doubling seven times yields The last entry of the table has ⁄ as one of its best rational approximations; i.e., there is no better