We know for a 2-dimensional surface there are two orthogonal principal directions at every point, where the principal curvatures $\kappa_1$ and $\kappa_2$ are the two ends of the curvature spectrum over all directions at that point. Gauss and mean curvatures are defined as $K=\kappa_1\kappa_2$ and $H=(\kappa_1+\kappa_2)/2$ respectively.
My question is why we use these two specific combination of principal curvatures to quantify a surface's deviation from being "flat" at that point? why not use for example $\kappa_1^2+\kappa_2^2$? A single number can never describe the curvedness in all directions, so what's the advantage of $K$ and $H$? More generally, what criterions are used to decide whether a definition of curvature is "good"?