Wikipedia tells me that:
Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane:
$K = \lim_{r \rightarrow 0} (2 \pi r - \mbox{C}(r)) \cdot \frac{3}{\pi r^3}$
Gaussian curvature is the limiting difference between the area of a geodesic circle and a circle in the plane:
$K = \lim_{r \rightarrow 0} (\pi r^2 - \mbox{A}(r)) \cdot \frac{12}{\pi r^4}$
Can anyone explain to me why we are dividing by the factors $\frac{3}{\pi r^3}$ and $\frac{12}{\pi r^4}$ respectively? I don't understand why we are dividing by these particular factors?
Many thanks in advance!
