Cone over a circle.

Question

Let K be a cone over a cirlce $C$ of Length L.That is $K=C\times[0,\infty)/(C,0)$ it is normal euclidean cone. In notation of Bridson Haefliger here is a link http://www.math.bgu.ac.il/~barakw/rigidity/bh.pdf. it is just $C_0(C)$. The metric is the usual metric over the cone in the book. Can anyone suggest a proof of the fact that the cone K is of non positive curvature iff $L\geq 2\pi$ and nonnegative curvature iff $L\leq 2\pi$.