Area of a circle Measurement of a Circle" (c. 260 BCE), compare the area enclosed by a circle to a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius. If the area of the circle is not equal to that of the triangle, then it must be either greater or less. We eliminate each of these by contradiction, leaving equality as the only possibility. We use regular polygons in the same way. Suppose that the area "C" enclosed by the circle is greater than the area "T" = ⁄"cr" of the triangle. Let "E" denote
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