The water in a cone is itself a cone, similar to the container. Drag the surface and watch the depth move by k, the top surface by k², the volume by k³.
Similar solids · working backwards
depthk
top surfacek2
volumek3
Length × k → Area × k2 → Volume × k3Only because the water is a cone similar to the container. In a cylinder, or a cone standing point up, the water is not similar and cube rooting fails.