  The atomic dipole polarizability, $\alpha$, and the van der Waals (vdW)
radius, $R_{\rm vdW}$, are two key quantities to describe vdW interactions
between atoms in molecules and materials. Until now, they have been determined
independently and separately from each other. Here, we derive the
quantum-mechanical relation $R_{\rm vdW} = const. \times\alpha^{1/7}$ which is
markedly different from the common assumption $R_{\rm vdW} \propto
\alpha^{1/3}$ based on a classical picture of hard-sphere atoms. As shown for
72 chemical elements between hydrogen and uranium, the obtained formula can be
used as a unified definition of the vdW radius solely in terms of the atomic
polarizability. For vdW-bonded heteronuclear dimers consisting of atoms $A$ and
$B$, the combination rule $\alpha = (\alpha_A + \alpha_B)/2$ provides a
remarkably accurate way to calculate their equilibrium interatomic distance.
The revealed scaling law allows to reduce the empiricism and improve the
accuracy of interatomic vdW potentials, at the same time suggesting the
existence of a non-trivial relation between length and volume in quantum
systems.
