Swapping a nine-inch round for a nine by thirteen is not a small change — the rectangle has nearly twice the floor area. Scaling by area is the right approach because area is what sets how deep the batter sits, and depth is what sets how it bakes.
Area, not diameter
The common mistake is scaling by the linear dimension. Going from an eight-inch round to a nine-inch round sounds like a 12 percent increase; it is actually 27 percent, because area goes with the square of the diameter.
An eight-inch round is 50 square inches. A nine-inch round is 64. A nine by thirteen rectangle is 117 — more than twice the eight-inch round.
Two eight-inch rounds is 100 square inches, which is why a two-layer eight-inch cake and a single nine by thirteen sheet use roughly the same amount of batter.
Depth is the hidden variable
Cake batter should fill a pan between half and two thirds. Below that it bakes too fast and the edges dry before the centre sets. Above it, the batter rises over the rim.
So the area ratio is the starting point, and then you check depth. If the scaled batter only fills the new pan a third of the way, the pan is too big for the recipe regardless of what the arithmetic says.
Pan height varies more than people notice. A standard round is two inches deep; a professional one is three. Same diameter, 50 percent more capacity.
Bake time does not scale with the multiplier
Heat reaches the centre of a cake by conduction from the edges and the base, so what governs bake time is the depth of the batter and the distance to the middle — not the total volume.
A deeper cake takes longer, sometimes much longer, and often needs a lower temperature so the outside does not set before the middle. A shallower sheet of the same volume bakes faster.
Start checking at the original time and use a skewer. Never trust a scaled recipe's time.
Eggs are the awkward ingredient
Most ingredients scale continuously. Eggs do not — you can use two or three, and a scale factor of 1.27 asks for 3.8.
The practical fix is to round to the nearest whole egg and adjust the liquid a little to compensate. A large egg is about 50 grams without the shell, roughly 30 of it white, so you can also weigh out a part egg if precision matters.
For a recipe that lives or dies on structure — a sponge, a soufflé, a chiffon — weigh the egg. For a butter cake, round and move on.
When to scale and when not to
Up to about 1.5 times, scaling by area works reliably for butter cakes, brownies and bar recipes.
Beyond that, the geometry of the batter changes enough that the bake is a different problem. Cheesecakes, anything with a custard set, and anything leavened only by whipped egg are the least forgiving.
For a large jump, making the recipe twice in the pan it was written for gives a better result than one scaled bake, and it is less risk.