Abstract: Let X be a Banach space, X₂ be a two dimensional subspace of X, and S(X) = {x in X : ||x|| = 1} be the unit sphere of X. The relationship between normal structure and the arc length in X is studied. Let R(X) = inf {l(S(X₂))- r(X₂): X₂ subseteq X }, where l(S(X₂)) is the circumference of S(X₂) and r(X₂})= sup {2(||x + y|| + ||x - y||): x, y in S(X₂)} is the least upper bound of the perimeters of the inscribed parallelogram of S(X₂). The main result is that R(X) > 0 implies X has uniform normal structure.