Additional iterations do not have simple closed forms, but the numerical approximations for ,
1, 2, 3, 4 (corresponding to 6-, 12-, 24-, 48-, and 96-gons) are

(9)

(10)

(11)

(12)

(13)

By taking
(a 96-gon) and using strict inequalities to convert irrational bounds to rational
bounds at each step, Archimedes obtained the slightly looser result

Miel, G. "Of Calculations Past and Present: The Archimedean Algorithm." Amer. Math. Monthly90, 17-35, 1983.Phillips,
G. M. "Archimedes in the Complex Plane." Amer. Math. Monthly91,
108-114, 1984.