Curvature
 

Reading Gauss biography I found this definition of "gaussian curvature"
In each point of a surface "S" exists a normal vector. Imagine that all normals are traced. Now in the center of a sphere (that can be anywhere with reference to the surface), with unit radio, imagine that all the radios parallel to the normals of the surface "S" are traced. The extremes of these radios make a curve "C" over the sphere of unit radio.
The area of the spheric surface inside "C" is defined as the gaussian curvature.

Now my question is the following: What does the length of the curve C over the unit sphere mean, in relation with the curvature of the surface S?
Maybe it has something to do with the "mean" curvature?