Question

How to find the centroid of a cone with radius r and height h?

Answer 1

If it's a right circular cone, I thought they were always one-third the way up from the base.

Answer 2

I'm trying to remember how to derive that though . . .

Answer 3

Thanks CliffSedge, but I meant it in terms of using integrals and all that mess.

Answer 4

does anyone know of a number that adds to four and multiplies to -3??

Answer 5

Right. . . It's been about a decade since I've done those. I think I remember that it's somewhat similar to using an integral to find an average value.

Answer 6

The centroid of a cone is located on the line segment that connects the apex to the centroid of the base.
For a solid cone, the centroid is 1/4 the distance from the base to the apex = h/4

Here's how we prove this.
The volume of cone is,
\[dV= \pi r(x)^2 dy\]
and its centroid is located at y
\[\frac {r(x)}{(h-y)}=\frac{r}{h}\]
\[r(x)=\frac{r(h-y)}{h}\]
now, \[dV= \frac { \pi r^2 (h-y)^2 }{h^2} dy\]

Integration \[y = \frac{ \int\limits y dV} {\int\limits dV}\]

\[\int\limits ydV = \frac{ \pi r^2 y(h-y)^2}{h^2} dy = \frac{ \pi r^2 h^2}{12}\]

\[\int\limits dV = \frac{ \pi r^2 (h-y)^2}{h^2} dy = \frac{ \pi r^2 h}{3}\]

Substitute in y, we get
\[ y = \frac{ \int\limits y dV} {\int\limits dV} = \left( \frac{ \pi r^2 h^2}{12} \right) \left( \frac{3}{ \pi r^2 h} \right) = \frac{h}{4}\]