If one has a triangle, the centroid is located 1/3 of its height from its longest edge, see: [triangle centroid][1], however for a cone it is instead located a quarter from this edge. I cant wrap my head around this, as a cone is simply a revolved triangle, and if one were to take a slice of the cone, to convert it into a triangle, then suddenly the centroid has moved. Why does this occur for cones but not for other shapes like spheres and cubes, where the centroid location remains constant for when slices are taken.
2026-02-23 19:22:26.1771874546
centroid of a cone in comparison to the centroid of a triangle
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1
If you double the height of a triangle (keeping the same angles) then its area increases by a factor of four. ($2^2$). So the section that you added to the bottom of the triangle to double its height has three times the area of the original triangle.
If you double the height of a cone, then its volume increases by eight. ($2^3$). This is because a cone is a three-dimensional object while a triangle is a two-dimensional object. Now, the section that you added to the bottom of the cone to double its height has seven times the volume of the original cone.
Because lower sections of the cone have proportionally more of the cone's volume (compared to the proportion of the area in the lower sections of the triangle), as you can see by thinking about doubling the cone's dimensions, the centroid is closer to the bottom of the cone.
This happens for cones and not spheres or cubes, because spheres and cubes have more symmetry than cones.