How much soil does a pot need?
Most pots are narrower at the base than the top, so a cylinder calculation over-orders soil.
Fill to buy
Enter both widths and get the real volume.
Worked example: a 16 in pot with a 12 in base
A typical large patio pot is 16 in across the top, 12 in across the base and 14 in tall. Treating it as a cylinder of its top width overstates the volume by 23%.
The formula for a tapered container is the frustum volume, (πh/3)(R² + Rr + r²) for a round pot and (h/3)(A² + Aa + a²) for a square one, where the capitals are the top dimensions and the lower-case letters the base. Measuring the base as well as the top is the difference between one bag and two.
Pot size in gallons is not a volume you can trust
Nursery pots are labelled in gallons, but a “5 gallon” pot usually holds about 3.5 to 4 US gallons of actual mix, because the trade gallon is a nominal size rather than a measurement. Weigh the arithmetic instead: measure the top, the base and the height, and calculate.
Refilling an existing pot
You rarely need a full refill. Roots and old mix occupy perhaps half the volume, and the mix compacts. A practical approach is to tip the old mix out, break it up, remove the top third and its roots, then refill with fresh mix and a slow-release fertiliser. Reusing mix for the same plant family is where disease builds up: tomatoes and potatoes especially should get fresh mix.