Converting a 10″ round recipe to a 9″ square pan
Batter scales with volume, not with width. A 10″ round pan holds about 11 cups filled to the brim, and a 9″ square pan holds 10 cups, so you need 9% less batter. Multiply every ingredient — including the eggs, which is the awkward one — by 0.91×.
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 1.3″ 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
A 10″ round recipe fills about one 9″ square at the usual two-thirds level — a clean split, which makes this an easy conversion to do without scaling anything.
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.