Salty & Clever
The Pan Calculator
The recipe wants a pan you do not own. Volume, depth and time for every common swap, plus the ones that do not work.
Assumptions to check
For straight-sided pans, the recipe factor preserves the original batter depth using base area. The separate fill check catches insufficient height. Shaped pans use capacity only. Fill fractions and time ranges are house estimates; timing shown is for the whole unscaled batch. Use the original recipe as the timing starting point after scaling to the original depth. Check actual internal dimensions and doneness.
Salty and Clever · the pan calculator
The recipe wants a pan you do not own.
Every chart on the internet will hand you a ratio and leave. The ratio is the easy half and it is not the half that ruins the cake. What ruins the cake is depth: the same batter in a wider pan is shallower, cooks faster and comes out dry, and in a narrower pan it is deeper, rises past the rim and burns at the edge before the middle sets.
This compares base area to preserve batter depth, then checks volume and usable height. Shaped pans use capacity because their depth cannot be inferred honestly from the stated diameter.
Pick two pans.
—of the recipe
The bar is the pan, side on. The vertical line is where this kind of batter should stop; anything past it is shaded, and anything past the end of the bar is on the oven floor.
| Pan | To the brim | Usable | Batter sits |
|---|
What to do about the time
If you would rather not scale the recipe
Why depth is the number, not diameter
Heat enters a cake from the outside and has to travel to the middle. Widen the pan and the journey gets shorter, so the middle sets sooner and the edges spend less time drying out while they wait. Deepen it and the journey gets longer, so the outside is well past done by the time the center stops being batter.
That is why swapping a nine-inch round for an eight-inch round is not a small change even though it sounds like one. The eight holds about a fifth less, so the same batter sits about a quarter deeper, and a quarter deeper is roughly seven extra minutes on a thirty-five minute cake — with the edge taking all of them.
It is also why the two directions fail differently. Too deep is a visible failure: it climbs the pan, spills, sinks in the middle, and you know. Too shallow is an invisible one. It looks perfect and it is dry, and most people blame the recipe.
The fill fractions this uses
A pan is never filled to the rim, and how far short you stop depends entirely on what is going in it. These are the fractions behind every number above.
| What it is | Fill to | Why |
|---|
Where the volumes come from
The rectangular and round pans are computed from their dimensions — length times width times depth, or pi times the radius squared times the depth. That is the volume to the actual brim, which runs a little above the figures printed in most pan charts, because those are quietly quoting a usable fill rather than a capacity.
The bundt, the tube pan, the pie plate and the muffin tin are not computed. They slope, they have a hole in the middle, or both, and geometry on a stated diameter would produce a confident wrong answer. Those four carry their published capacity instead, and the table says so.
Measure your own pan across the inside of the top rim, not the outside. A pan sold as nine inches is often eight and three quarters where it counts, and on a round pan that is six percent of the volume gone.
The whole argument, written out: an 8-inch pan is not a smaller 9-inch pan.
The swaps that are actually fine
Four of them need no arithmetic at all, and almost every chart buries them under forty rows that do.
An eight-inch square and a nine-inch round are within half a percent of each other — 128 cubic inches against 127. A nine-inch square and a ten-inch round are within three. A nine by thirteen and a half sheet pan hold exactly the same, 234 cubic inches each, because thirteen times eighteen by one inch and nine by thirteen by two inches happen to land on the same number; two quarter sheets are that same volume again.
What is not a clean swap, and gets repeated as though it were: two eight-inch rounds do not equal a nine by thirteen. They are fourteen percent short, which is enough to notice. Two nine-inch rounds overshoot it by nine, which is the closer of the two and still not nothing.
The one that is never fine
Do not put a batter meant for a tube pan into a solid one. An angel food or a chiffon is engineered around the hole in the middle: it is what lets a very deep, very wet foam cook through before the outside sets, and it is what the cake hangs from while it cools upside down. Without it the center never sets and the whole thing collapses into the pan as it cools. That is a different dish, not a deeper one.
Where this goes next
Changing the pan is one kind of scaling. Changing the number of people is another, and it breaks in different places — the pan and the oven and the afternoon do not multiply the way the ingredient list does.
Free, next door
The Scaling Calculator takes a recipe up or down for a crowd and names what must not be multiplied. The Salt Calculator handles the seasoning half, which is the thing that goes wrong first when the quantity moves.
The one you pay for
The Field Manual is these tables printed — pan volumes, fill fractions, salt percentages, hold times and temperature bands — sized for a cupboard door, for the evening when your hands are covered in batter and the laptop is in the other room.
Also free: every number on one page, the board calculator, the bar calculator, the oven traffic calculator, the roast clock, all of them.
How this calculator works, and where every number in it comes from
Every pan-substitution chart online gives you a bare ratio: a 9 by 13 is about twice an 8 by 8, go ahead. That is true about volume and silent about the thing that actually decides whether the swap works, which is how deep the batter sits and how long the heat takes to reach the middle of it.
The volume formula, written out
Volume is geometry, to the brim. A round pan is pi multiplied by the radius squared, multiplied by the height. A rectangular pan is length multiplied by width multiplied by height. One cup is 14.4375 cubic inches.
Those figures run above the capacities printed in most pan charts, and deliberately so, because published charts quietly quote a usable fill rather than the actual vessel. Stating the brim volume and the fill fraction separately is more honest than folding one into the other and not saying.
Four pans carry a published capacity instead of a computed one, because geometry on a stated diameter would lie about them: a bundt at 12 cups, a tube pan at 16, a pie plate at 4, a muffin tin at 6.
The fill fractions, which are the real answer
What fraction of the pan the batter should occupy depends entirely on what it is going to do in there. Leavened cake fills to 0.60, because it has to climb. Brownies fill to 0.75. Quick bread fills to 0.65. Yeast bread fills to 0.50, because it climbs further than anything. A casserole fills to 0.85. A cheesecake fills to 0.90, because it does not rise and it is baked gently.
Then time moves with depth raised to the power of 0.8, shown as a range starting at the low end, because an under-checked cake is recoverable and an over-baked one is not.
Two worked examples
An 8-inch round to a 9-inch round, leavened cake. The 8 is pi multiplied by 16 multiplied by 2 inches, so about 100 cubic inches, or 7 cups. The 9 is pi multiplied by 20.25 multiplied by 2, about 127 cubic inches, or 8.8 cups. The batter that filled the 8-inch to 0.60 fills the 9-inch to 0.47, so it sits shallower, bakes faster, and comes out drier with a paler top. Take five to eight minutes off and check early.
An 8-inch square to a 9-inch round. The square is 64 square inches of floor, the round is 63.6. They match within one percent, which makes this the one substitution nobody needs to think about and everybody does.
The editorial point, which is why the direction matters
The two failure directions do not look the same and that is the whole trap.
Too deep is visible. It climbs, it spills, it sinks in the middle, and you know immediately what went wrong.
Too shallow is invisible. It looks perfect. The edges are a little firm and the crumb is a little dry, and the cook blames the recipe, the flour, or themselves, and makes the same substitution again next time.
Assumptions, and when this breaks
Dimensions are assumed to be measured across the top inside edge, which is how pans are labeled and is not how they are shaped. A pan with sloped sides holds less than its stated top diameter implies, and a dark pan bakes faster than a light one at the same size, neither of which the arithmetic can see.
The time exponent is an approximation fitted across ordinary domestic depths. It is not reliable for anything very shallow, like a jelly roll, or very deep, like a tall tiered cake, and it does not know about water baths at all.
It breaks completely on one substitution: never put a tube-pan batter into a solid pan. The hole is not decorative. It is what cooks the middle and what the cake hangs from while it cools, and a tube batter in a solid pan is raw in the center and collapses on the way out.
Every capacity and fill fraction above is listed on Every Number This Site Runs On. For the savory version of the same physics, where the constraint is evaporation and browning rather than depth, the piece is every recipe assumes a pan diameter.
Where this goes next
You have the number. Now use it.
Put the number to work
Vacuum Sealers: Worth It for Freezers, Sous Vide, and Bulk Buying – Not Everyone Else
Vacuum sealers can reduce freezer burn, improve bulk storage, and simplify sous vide prep. They also add plastic, bag costs, counter space, and a cleaning problem when liquids reach the sealing channel.
Tools + finds
Vegetable Choppers: Faster Prep or Another Plastic Thing to Wash?
Air Fryers: What They Do Better Than an Oven – and What They Don’t
The next tool
The Salt Calculator
Grams of salt for what you are actually cooking, by the brand in your cupboard.The Scaling Calculator
Take a recipe from four to fourteen without scaling the salt and the pan along with it.Every Number This Site Runs On
The reference sheet the calculators run on, in one place.When the free answer is not enough
Kitchen system · $15