A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter $12$ and altitude $12$, and the axes of the cylinder and cone coincide. Find the radius of the cylinder.

1. Let $r$ be the cylinder radius. Its height is $2r$. At height $2r$ above the cone's base, the cone's radius is $6\left(1-\frac{2r}{12}\right)=6-r$.
2. Tangency requires $r=6-r$, so $r=3$.