A while ago, my parents’ neighbor asked a good question: For a tapered barrel, is there an equation for the volume versus height? Here’s what I came up with.

When comparing the volume ratio for specific barrels, the following fill curves occur:

Now holding a constant maximum radius, the following absolute volume ratios (compared to the straight sided barrel) occur:

Rules of thumb:

• The fill-volume ratio is approximately (±15%) linear with height when the upper to lower radius ratio is within 2:1.
• Half the volume of a cone is at the 20% height level.

The final recommendation was to fill the barrel with a known volume (gallons) and mark the side.