jmc

First, we label the diagram as shown below:



Both right triangles are 30-60-90 triangles. Therefore, the length of the shorter leg in each triangle is half the hypotenuse, and the length of the longer leg is $\sqrt{3}$ times the length of the shorter leg. We apply these facts to each triangle, starting with $△AOB$ and working clockwise.

From $△AOB$, we find $AB=AO/2=8$ and $BO=AB\sqrt{3}=8\sqrt{3}$.

From $△BOC$, we find $BC=BO/2=4\sqrt{3}$ and $CO=BC\sqrt{3}=4\sqrt{3}\cdot \sqrt{3}=\boxed{12}$.