An edge of a cube measures $r\text{cm}$. If the largest possible right circular cone is cut out of this cube, then the volume of the cone (in ${\text{cm}}^3$) is

For the largest possible right circular cone to be carved out of a cube of side length $r$:The diameter of the cone's base must equal the edge of the cube: $d=r⟹\text{radius }R=\frac{r}{2}$ The height of the cone must equal the edge of the cube: $h=r$ Calculate the volume of the cone ($V$):
$$V=\frac{1}{3}\pi R^{2}h$$
$$V=\frac{1}{3}\pi {\left(\frac{r}{2}\right)}^2(r)$$
$$V=\frac{1}{3}\pi \left(\frac{r^2}{4}\right)(r)=\frac{1}{12}\pi r^3$$