In order to study the relation between heights of unit equilateral triangles and geometric properties of the underlying space

a new geometric constant h(X) is introduced via geometric method.The lower bound of h(X)is presented.It is proved that a Banach space X is uniformly nonsquare if and only if the inequality h(X)1/2 holds

and that a Banach space X has the fixed point property for nonexpansive mappings whenever the inequality h(X)1/2 holds.