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 formula_53. The calculations Archimedes used to approximate the area numerically were laborious, and he stopped with a polygon of 96 sides. A faster method uses ideas of Willebrord Snell ("Cyclometricus", 1621), further developed by Christiaan Huygens ("De Circuli Magnitudine Inventa", 1654), described in . Given a circle, let "u" be the perimeter of an inscribed regular "n-"gon, and let "U" be the perimeter of a circumscribed regular "n-"gon. Then "u" and "U" are lower and upper bounds for the circumference of the circle that become sharper and sharper as "n" increases, and their average ("u" + "U")/2 is an especially