The ratio of the areas of the triangle and rectangle below is $1:2$.
So why isn't the ratio of volumes of a cone and the smallest cylinder that contains it $1:2$? If each "slice" has a $1:2$ ratio then shouldn't the volumes have a $1:2$ ratio as opposed to a $1:3$ ratio?
Think about:
In your triangle picture, the ratio of triangle width to square width at varying heights decreases linearly from $1$ to $0$ (as you go from bottom of the figure to top). However, the ratio of cone cross-sectional area to cylinder cross-sectional area goes from $1$ to $0$ quadratically (because of area being proportional to square of radius).