Find the centroid of a right circular cone with base
Find the centroid of a right circular cone with base radius $r$ and height $h$.
Explanation of Method
Place the apex at the origin with the axis along $z$; by symmetry the centroid lies on the axis, so compute $\bar z=M_{xy}/V$ where $M_{xy}=\iiint z\,dV$, using cylindrical coordinates with cross-sectional radius $\rho(z)=rz/h$.
Step 1: Volume
\[V=\frac13\pi r^2h\]
Step 2: Moment about the base plane through the apex
\[M_{xy}=\int_0^h\int_0^{2\pi}\int_0^{rz/h}z\rho\,d\rho\,d\theta\,dz = 2\pi\int_0^hz\cdot\frac{r^2z^2}{2h^2}\,dz=\frac{\pi r^2}{h^2}\cdot\frac{h^4}4=\frac{\pi r^2h^2}4\]
Step 3: Centroid height from the apex
\[\bar z=\frac{M_{xy}}V=\frac{\pi r^2h^2/4}{\pi r^2h/3}=\frac{3h}4\]
Step 4: Result
The centroid lies on the axis, at $\dfrac{3h}4$ from the apex, equivalently $\dfrac h4$ from the base.
\[\boxed{\text{Centroid at }\frac h4\text{ above the base (}\frac{3h}4\text{ from the apex)}}\]
Result
Final answer:
\[\boxed{\text{Centroid lies on the axis, } \dfrac{h}{4} \text{ from the base } \left(\dfrac{3h}{4}\text{ from the apex}\right)}\]