The centroid makes sense intuitively when you think about it as a center of mass. Since it's the average position, for every portion that's father away there is, in a sense, a portion that's closer - the portion being equal either insofar as it is larger and closer to the center or smaller and father away, and since the size/volume of a ring increase linearly with distance in an intuitive manner, you can justify to yourself that you should be using the centroid without any formal calculus, just by considering the "weight" of a ring.
This is an excellent explanation. If we imagine the triangle being swept about in a circle, the points in the "outer" section (forming the circle at edge of the base our our cone) will cover more distance the points in the "inner" section (the points near the center line of our cone that we're revolving around). The distance moved by the centroid, then, is the average distance a point travels.
