I've learnt that the capacitance $C$ of a parallel plate capacitor is given by: $$C=\frac{A\epsilon_0}{d}$$ where, $A$ is the area of cross section of the plates and $d$ is the separation between the two plates. When the space in between the two plates is filled by a dielectric of dielectric constant $K$ the new capacitance is given by: $$C'=KC=K\frac{A\epsilon_0}{d}$$ When the space between the plates is filled by a dielectric of dielectric constant $K$, the capacitance is increased by a factor of $K$. Is this applicable for all types of capacitors (spherical, cylindrical, etc.)? If yes, what is the reason behind this fact? For example, the capacitance of a spherical capacitor is given by:
$$C=\frac{4\pi\epsilon_0 r_1 r_2}{r_2-r_1}$$
where, $r_1$ and $r_2$ are the radii of inner and outer metallic shells respectively. If we fill the entire region between the capacitor with a dielectric of dielectric constant $K$ will the resultant capacitance be given by:
$$C'=KC=K\frac{4\pi\epsilon_0 r_1 r_2}{r_2-r_1}\ \ ?$$