11.225 A cone lies on a plane and rolls along it, rotating around its fixed vertex. The height of the cone and its slant height are equal to $h$ and $l$. Calculate the area of the surface described by the height of the cone.
## Solution.

Let $\triangle A B F$ be the axial section of the given cone,

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-276.jpg?height=293&width=362&top_left_y=724&top_left_x=90)
$l=A F=B F$. Since the angle $A F K$ between the height $K F$ of the given cone and the plane on which it rolls does not change, $K F$ describes a conical surface with the generatrix $h=K F$ and the axis

FM. $\quad \triangle F K M \sim \triangle F K B \Rightarrow \frac{h}{l}=\frac{K M}{h} \Rightarrow K M=\frac{h^{2}}{l}$ and the area of the surface described by $K F$ is $\pi \cdot K M \cdot K F=\frac{\pi h^{3}}{l}$.

Answer: $\frac{\pi h^{3}}{l}$.