If the side $AB$ of an equilateral triangle $ABC$ is equal to $12\text{cm}$, calculate the length (in $\text{cm}$) of $AO$, where $O$ is the centroid of the triangle.

In an equilateral triangle with side $a=12\text{cm}$, the length of the median (which is also the altitude $h$) is given by:
$$h=\frac{\sqrt{3}}{2}a=\frac{\sqrt{3}}{2}\times 12=6\sqrt{3}\text{cm}$$
The centroid $O$ divides the median/altitude from any vertex in the ratio $2:1$. The distance from the vertex $A$ to the centroid $O$ is $\frac{2}{3}$ of the total height:
$$AO=\frac{2}{3}h=\frac{2}{3}\times 6\sqrt{3}=4\sqrt{3}\text{cm}$$