Why is there a half in the definition of mean curvature?

Question

Using the language of basic geometry, I have a pretty naive question. Given a surface embedded in $\mathbb{R}^3$, modulo some details, we consider the shape operator and its eigenvalues $\kappa_1$ and $\kappa_2$ are called the principal curvature. Then the mean curvature $H$ is defined to be $$H = \frac{\kappa_1 + \kappa_2}{2} = \frac{1}{2}\cdot(\text{trace of the shape operator})$$ My question is why do we have that $1/2$ factor in the definition? My initial instinct was because then $H$ is the minimum value of the characteristic polynomial of the shape operator. But that's not really a reason, so I was wondering if someone has any thoughts on this and maybe a better reason.