Cylinder inscribed in a cone

Text

A cylinder of height $6$ cm is inscribed in a cone of base radius $12$ cm and height $18$ cm, with one base resting on the cone’s base. Find the cylinder’s radius, its volume and its lateral surface.

Solution

By similarity, the cone radius at height $y$ from the base is $R\left(1-\dfrac yH\right)$; at $y=6$: $r=12\left(1-\frac{6}{18}\right)=12\cdot\frac23=8\ \text{cm}.$ Cylinder volume: $V=\pi r^2 h=\pi\cdot64\cdot6=384\pi\approx1206.37$ cm$^3$. Lateral surface: $S_\ell=2\pi r h=2\pi\cdot8\cdot6=96\pi\approx301.59$ cm$^2$. $\boxed{r=8\ \text{cm};\ V=384\pi\approx1206.37\ \text{cm}^3;\ S_\ell=96\pi\approx301.59\ \text{cm}^2}$