I\'m looking for more of an analytical answer here rather than a mathmaticle one

Question

I'm looking for more of an analytical answer here rather than a mathmaticle one.

Why is the centroid of a triangle located at (1/3) of the height, while the centroid of a cone is located at (1/4) of the height? I would've assumed that since the cone is just the integral of the triangular area from 0 to 2pi the centroid would be located at the same height as its crossectional shape.

Again I know that the integrals work out to give 1/3 and 1/4, I'm jsut looking for an explaination as to why this is so.

Explanation / Answer

if you think about the mass distribution in the triangle vs a cone, you can see that there is much more mass in the larger radius section of the cone.volume is proportional to the radius squared in the cone, while a triangle is based one just the radius. this would explain why the cone's mass is centered closer to its flat surface. another way to say it is that the triangle's mass increases linearly as the base width increases, but the cone's increases as a square of its base radius.

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