The ratio between the radius of the base and the height of a cylinder is 2 : 3.

Let the radius of the base and the height of the cylinder be $r$ and $h$, respectively.

We have,

$r: h=2: 3$ i. e. $\frac{r}{h}=\frac{2}{3}$

or $h=\frac{3 r}{2} \quad \ldots$ (i)

As,

Volume of the cylinder $=12936 \mathrm{~cm}^{3}$

$\Rightarrow \pi r^{2} h=12936$

$\Rightarrow \frac{22}{7} \times r^{2} \times \frac{3 r}{2}=12936 \quad[$ Using $(\mathrm{i})]$

$\Rightarrow \frac{33}{7} \times r^{3}=12936$

$\Rightarrow r^{3}=12936 \times \frac{7}{33}$

$\Rightarrow r^{3}=2744$

$\Rightarrow r=\sqrt[3]{2744}$

$\therefore r=14 \mathrm{~cm}$

So, the radius of the base of the cylinder is 14 cm.