Derive an expression for the capacitance of a parallel plate capacitor?

Two plates of same area (A) are seperated by a small distance (d) i.e, considered as a parallel plate capacitor. The space between the plates is filled with vaccum (or) air
Let 'q' is charge on each plate $\mathrm{\sigma }$ is surface charge density on either plate
$\mathrm{\sigma }=\frac{\mathrm{q}}{\mathrm{A}}$ 
The electric field between the plates is uniform and its magnitude is given by
$\begin{array}{c}\mathrm{E}=\frac{\mathrm{\sigma }}{{\mathrm{ϵ}}_0};\\ \mathrm{E}=\frac{\mathrm{q}}{{\mathrm{Aϵ}}_0}\end{array}$

for uniform electric field the potential difference between the plates is given by V = Ed
$\begin{array}{l}\mathrm{V}=\frac{\mathrm{q}}{\mathrm{A}{\in }_0}.\mathrm{d}\\ \frac{\mathrm{q}}{\mathrm{V}}=\frac{{\in }_0\mathrm{A}}{\mathrm{d}}\text{ here }\frac{\mathrm{q}}{\mathrm{V}}=\mathrm{c}\\ \mathrm{C}=\frac{{\in }_0\mathrm{A}}{\mathrm{d}}\text{ in medium }\\ \mathrm{C}=\frac{\mathrm{\epsilon A}}{\mathrm{d}};\mathrm{C}=\frac{\mathrm{K}{\in }_0\mathrm{A}}{\mathrm{d}}\end{array}$
where K is relative permitivity of medium