Area of a circle to eight sides and so on. For a polygon with 2"n" sides, the parallelogram will have a base of length "ns", and a height "h". As the number of sides increases, the length of the parallelogram base approaches half the circle circumference, and its height approaches the circle radius. In the limit, the parallelogram becomes a rectangle with width "r" and height "r". There are various equivalent definitions of the constant π. The conventional definition in pre-calculus geometry is the ratio of the circumference of a circle to its diameter: However, because the circumference of a circle is not a
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