Infinite Powers: The Story of Calculus - The Language of the Universe

by Steven H. Strogatz

He began with a hexagon because it was a convenient base camp from which to embark on the more arduous calculations ahead. The advantage of the hexagon was that he could easily calculate its perimeter, the total length around the hexagon. It’s six times the radius of the circle. Why six? Because the hexagon contains six equilateral triangles, each side of which equals the circle’s radius. Six of the triangle’s sides make up the perimeter of the hexagon. So 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 Thus the hexagon argument demonstrates π > 3.