How do I substitute a pan of a different size?

Compare base areas first: square/rectangles are length × width, rounds are π × (diameter ÷ 2)². If areas match within about 10%, bake the recipe as-is — King Arthur Baking publishes 9-inch round (64 in²) and 8-inch square (64 in²) as a direct swap. If areas differ, scale the recipe by the area (or volume) ratio and adjust the clock: thinner bakes finish faster, deeper bakes need longer at a slightly lower temperature.

What's the formula for a round pan equivalent to a square?

Multiply the square's side by 2 ÷ √π (about 1.128) for the round with the same base area; multiply a round's diameter by √π ÷ 2 (about 0.886) for the equivalent square. That's how you get the classic 8-inch square ↔ 9-inch round pair, which King Arthur lists as interchangeable in any recipe.

Can I halve a 9×13 recipe into an 8-inch square?

Almost exactly: a 9×13 pan is 117 in² and an 8-inch square is 64 in² — 55% of the area, so a half recipe fits with a hair of extra depth. King Arthur's own example runs the other direction: increase a 9-inch-square recipe by about 50% for a 9×13, or bake it as-is in an 11×7 (77 in²) for slightly thicker bars.

Why does batter depth matter more than volume?

Because doneness is a race between the edges (which set fast against hot metal) and the centre (which heats slowest). Double the depth and the centre's distance to heat roughly doubles while the top browns on the same schedule — the standard fixes are a lower temperature and a longer bake. The same volume in a wider pan bakes shallower and faster, risking a dry edge before the middle sets.

How do I scale eggs when scaling a recipe?

Multiply the egg count by the same scale factor, then round to the nearest half egg. When you need the half, beat an extra egg and measure out about 2 tablespoons (30 ml). If the factor is between roughly 0.9 and 1.1, leave the eggs alone and scale everything else.

When is a pan NOT substitutable?

When the pan is part of the recipe's physics: King Arthur's guide names 9-inch springform pans (3-inch walls built for cheesecakes) and 10×15 jelly-roll pans for sponge rolls as non-substitutable, and advises against foam cakes like angel food in Bundt pans, whose designs are not meant to be climbed by delicate batters.

Pans

Pan substitution math explained: area, volume and batter depth

Nov 16, 2026 · ~8 min read

Every pan swap is one of three calculations wearing different clothes: compare areas when the batter will be the same depth, compare volumes when it will not, and check batter depth when the bake time is at stake. This article does all three sums in the open, with numbers that match King Arthur Baking's published pan-swap guide — because the math is not a secret, it is three formulas.

Key takeaways

Area formulas: squares/rectangles = L × W; rounds = π × (d⁄2)². Both live in King Arthur's published table (8″ sq = 64 in², 9″ sq = 81, 11×7 = 77, 9×13 = 117, 9″ round ≈ 64, 8″ round = 50).
Round↔square: equivalent square side = diameter × 0.886 (9″ round → 7.98″ ≈ 8″ square — the published direct swap).
Within ±10% area: bake as-is. Deeper or shallower by more than that: scale the recipe by the ratio and adjust time, not just temp.
Eggs round to halves — beat an extra egg and use 2 Tbsp when you need the half.

Step 1: compare areas when depth can stay the same

Area is the right measure when the pans are the same depth and the question is “will my batter spread the same?” — brownies, blondies, traybakes, pastry. King Arthur's guide is built on this comparison and “assumes a pan depth of 2 inches”:

Pan	Area	Notes from King Arthur's guide
8″ round	50 in²	—
8″ square	64 in²	Direct swap for 9″ round
9″ round	≈64 in²	“can be substituted for one another in any recipe”
11″ × 7″	77 in²	Stand-in for a missing 9″ square (bars slightly thicker)
9″ square	81 in²	Scale recipe ×1.5 for a 9×13
9″ × 13″	117 in²	Halving 117 → 58.5 in²: two 9″ rounds ≈ +10%, two 8″ rounds ≈ −9%
10″ × 15″ jelly roll	150 in²	≈ one 9″ square + one 8″ square (145 in²) for bars

The round↔square formula falls out of setting the areas equal: side × side = π × (d⁄2)², so side = 0.886 × d. Nine-inch round: 7.98 — call it an 8-inch square. The pan converter runs this both directions for any dimension.

Step 2: compare volumes when depth changes

Volume is the right measure when the shapes differ in kind — a round into a loaf tin, a slab into a Bundt — because the batter depth will move even when the total amount stays put. The geometry is school-level: a cylinder for rounds (πr² × depth), a box for squares, rectangles and loaves (L × W × depth). Divide cubic inches by 14.4375 to get cups (1 cup = 240 ml).

Then apply the fill rule: batter = ⅔ of brim, the practice stated on Sally's Baking Addiction's pan chart. A 9×5 loaf is 9.3 cups to the brim, 6.2 of batter; a 9″ round is 8.8 brim, 5.9 batter — close enough that a round-cake recipe fits the loaf tin, but the depth changes, which brings us to step 3.

Step 3: respect batter depth

Doneness is a race: the edges set against hot metal while the centre heats from conduction alone. Depth stretches the centre's commute. Double the depth and you do not double the time — you lower the temperature and add much more of it. That is why King Arthur's guide, moving a standard cake into a 10-cup Bundt, adds about 30% more baking time (15–20% in a 12-cup), and why a shallow traybake in a too-big pan dries at the edges before the middle sets.

Same batter, deeper pan: drop ~15°C / 25°F, add time generously, tent with foil once it browns.
Same batter, shallower pan: check 5–10 minutes early; edges will beat the middle to done.
⅔ is the ceiling: if the batter line sits above ⅔ in the new pan, some of it belongs in a second pan, not in the oven's cleanup future.

Step 4: scale the recipe — and the eggs

Scale factor = new capacity ÷ old capacity (use the batter numbers). A recipe written for two 8″ rounds (11.8 cups batter) in one 9×13 (10.8) gives a factor of 0.92 — bake as-is for a slightly taller slab. A 9″ square recipe in a 9×13 gives 117⁄81 = 1.44, matching King Arthur's “increase by about 50%”.

Eggs are the one ingredient that refuses to scale smoothly. Multiply, then round to the nearest half; for the half, beat a second egg and pour about 2 tablespoons (30 ml). Within ±10% of ×1, leave the eggs entirely — a 5% egg error is invisible, a broken custard is not.

When not to substitute at all

Some pans are structural: 3-inch-wall springforms for cheesecake, 10×15 jelly-roll pans whose rim the sponge needs, tube pans whose centre column angel food climbs. King Arthur's guide names all three. Foam cakes also fail in fluted Bundts — the batter cannot climb the folds. If the pan is in the recipe's physics, no factor rescues it.

Last verified: 2026-11-16 — sources: King Arthur Baking (alternative pan sizes guide, recipe success guide), Sally's Baking Addiction (pan chart), fetched live on that date. Ready-reckoned rows: the pan volume chart.