Converting an 8″ square pan recipe to a 9×13″ pan
Batter scales with volume, not with width. An 8″ square pan holds about 8 cups filled to the brim, and a 9×13″ pan holds 14 cups, so you need 75% more batter. Multiply every ingredient — including the eggs, which is the awkward one — by 1¾×.
The number people miss is depth. Two pans can hold the same volume and still bake completely differently, because what matters is how thick the batter sits: volume divided by bake surface. Here that moves from about 1.2″ to 1.2″. Batter depth is similar in both pans. Bake time should be close to the original; start testing 5 minutes early.
How many pans one batch fills
Going this way, one 8″ square recipe doesn't fill a single 9×13″ pan — it comes to about 57% of one, so you'd get a noticeably shallow bake. Scale the recipe up by 1¾× instead and fill the pan properly. A cake baked in a pan that's too big for it dries at the edges before the centre has any height to it.
A caution worth reading
Volume conversions hold well for cakes, brownies and quick breads — anything poured. They hold less well for recipes where structure comes from the pan itself: a delicate chiffon or angel food climbs the ungreased side of a tube pan, and a loaf's shape is part of how it bakes. For those, scale the ingredients by all means, but expect the texture to shift, and use the skewer rather than the recipe's clock.