Area of a circle as follows. Draw a perpendicular from the center to the midpoint of a side of the polygon; its length, "h", is less than the circle radius. Also, let each side of the polygon have length "s"; then the sum of the sides, "ns", is less than the circle circumference. The polygon area consists of "n" equal triangles with height "h" and base "s", thus equals ⁄"nhs". But since "h" < "r" and "ns" < "c", the polygon area must be less than the triangle area, ⁄"cr", a contradiction. Therefore, our supposition that "C" might be greater than "T" must be
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