Diameter is a length, volume is a cube
Every pot on a shelf is labelled with a diameter. Every bag of mix is sold by volume. The trip between those two numbers passes through a cube, and that is where the estimating goes wrong.
Take the defaults on this page. A 6 inch pot tapering to 4.5 at the base, 5.5 inches deep, holds 119.9 cubic inches brim full — about 2.08 quarts. An 8 inch pot tapering to 6, 7 inches deep, holds 271.2 cubic inches, or 4.70 quarts. That is 2.26 times the volume for what the labels describe as a 1.33 times step in diameter. Cube 1.33 and you get 2.37, which is close to the 2.26 the real tapered shapes give, and the small difference is only because the two pots are not exactly the same proportions.
Run it the other way and the same fact is more useful. If you know one pot holds 2.26 times another, the dimension ratio is the cube root, 1.31 — so the second pot is about 31 percent bigger in every direction. Two steps of that size is not twice the mix. It is 5.1 times.
The root ball is not free space
The new pot does not need filling. It needs filling around something. At the defaults the root ball measures 5 inches across and 4.5 deep, which as a cylinder is 88.4 cubic inches, and the new pot filled to 90 percent of its height holds 236.9. The mix you actually buy is the difference, 148.5 cubic inches or 2.57 quarts, which is 37 percent less than the pot volume suggests.
The cylinder is a deliberate overestimate of a shape that is really a tapered plug, and overestimating the ball underestimates the mix. If precision matters, the displacement mode is the honest one: fill a jug, note the level, lower the ball in, note it again, and enter the difference in fluid ounces. That measures the shape you have rather than the shape it approximately resembles.
Fill level is not the same fraction as fill volume
Filling a tapered pot to 90 percent of its height gives 87.3 percent of its volume, not 90. The slice you leave empty is the top slice, and in a pot that gets wider as it rises the top slice is the fattest one. On the default new pot that gap is 34.3 cubic inches, about 0.59 quarts — small on one plant, and a bag on twenty.
In a straight-sided container the two percentages are identical, which is why the effect goes unnoticed until someone moves from nursery cans to tapered decorative pots and finds the mix running out early.
What the depth figures are for
When the root ball is entered by measurement, the page reports two clearances: the mix under the ball and the gap around it at the fill line. At the defaults that is 1.8 inches underneath and 1.4 inches each side. Those numbers exist so you can check them against the pot before you start, because the moment you discover the ball sits proud of the rim is always the moment your hands are full.
The gap is measured at the fill line, not at the rim. In a tapered pot the widest point of the mix is the surface, so the gap at any lower point is smaller than the figure shown, and the ball may touch the sides well before it reaches the bottom.
What this page will not tell you
It will not tell you whether to step up a pot size, when, or by how much. It will not recommend a mix, a fill level or a watering. Those are questions about a specific plant in a specific room, and they belong to whoever grows it — a nursery, an extension service, or your own record of what has worked. The arithmetic here is about containers.
Two physical cautions worth naming and no more. A watered pot is heavy, and a large one on a windowsill, a plant stand or a wall shelf is a load that furniture ratings cover only when someone has checked them. And potting media and the water standing under a pot both grow organisms; for most people that is unremarkable, and for anyone whose immune system is compromised it is a conversation with a doctor rather than with a calculator.
Questions people ask
Why does the calculator not accept a pot size in gallons?
Because a nursery gallon is a trade size rather than a measurement. Containers sold as one gallon commonly hold rather less than 231 cubic inches, the shortfall varies between manufacturers and between moulds, and the discrepancy compounds when you are subtracting a root ball from it. Measuring the inside of the pot you have removes the guess entirely, and a tape measure across the rim and down the inside takes about ten seconds. If you do want to work from nominal sizes and a fill percentage, the growing media and container volume calculator handles that case and reports how much the trade size is likely overstating.
The root ball is not a cylinder. Does that matter?
It matters in one direction: a cylinder is larger than the tapered, slightly rounded plug that actually comes out of a pot, so entering diameter and height overstates the ball and understates the mix you need. The error is usually somewhere between ten and twenty-five percent of the ball volume, which on the default numbers is roughly nine to twenty-two cubic inches, or up to about 0.4 quarts. If that matters, use the displacement mode instead and measure it in a jug. If you are buying by the bag anyway, the cylinder is the safe direction to be wrong in only if you round the bag count up.
What fill percentage should I use?
That is your call and the page does not make it. What the field is for is making the choice explicit and then showing you what it costs in volume, because the two are not proportional in a tapered pot: filling to 90 percent of the height gives 87.3 percent of the volume, and dropping to 80 percent of the height gives about 75 percent. If you are matching an existing pot, measure how far below the rim the mix currently sits and divide by the inside height.
Can I use this for a pot with no drainage hole, or one inside a cachepot?
The volume arithmetic is the same either way, because it only describes the space inside the container you measure. What changes is where water goes, and that is outside what this page calculates. If a growing pot sits inside a decorative outer pot, measure the growing pot for the mix figures — the outer one only matters for whether the inner one fits, and for the runoff volume it has to hold, which the pot watering calculator works out.
Why does the page show the volume of one more step up?
Because the compounding is the whole point and it is invisible on a shelf. If the step you entered is 2.26 times the volume, the next step of the same proportion is 5.1 times the pot you started from. People plan a sequence of repots in their head as a series of small moves, and the geometry makes it a series that doubles. Seeing the third number written out tends to change how much mix gets bought and how many shelves get cleared.