the perimeter equals six times the radius; in symbols, p = 6r. Then, since the circle’s circumference C is longer than the hexagon’s perimeter p, we must have C > 6r. This argument gave Archimedes a lower bound on what we would call pi, written as the Greek letter π and defined as the ratio of the circumference to the diameter of the circle. Since the diameter d equals 2r, the inequality C > 6r implies π = C / d = C / 2r > 6r / 2r = 3. Thus the hexagon argument demonstrates π > 3.
