Line segment $\overline{AB}$ is a diameter of a circle with $AB=36$. As point $C$ moves around the circle, the centroid of $\triangle ABC$ traces out a closed curve. To the nearest positive integer, what is the area of the region bounded by this curve?

1. Let $O$ be the center of the circle, the midpoint of $AB$. The centroid $G$ of $\triangle ABC$ lies on the median $CO$ with $OG=\frac13OC$, so as $C$ goes around the circle, $G$ traces a circle centered at $O$ with radius $\frac13$ of the radius of the given circle.
2. Radius here is $\frac13\cdot18=6$, so the area is $36\pi\approx\boxed{113}$.