Circumference pi, is represented by the Greek letter . The first few decimal digits of the numerical value of are 3.141592653589793 ... Pi is defined as the ratio of a circle's circumference to its diameter : Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference: The use of the mathematical constant is ubiquitous in mathematics, engineering, and science. In "Measurement of a Circle" written circa 250 BCE, Archimedes showed that this ratio (, since he did not use the name ) was greater than 3 but less
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