Converting a 9″ square recipe to a 9″ round pan
Batter scales with volume, not with width. A 9″ square pan holds about 10 cups filled to the brim, and a 9″ round pan holds 8 cups, so you need 20% less batter. Multiply every ingredient — including the eggs, which is the awkward one — by 0.80×.
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
A 9″ square recipe comes to a little over one 9″ round — about 1¼ pans' worth — so a single pan would be overfilled. Scale the recipe by 0.80× to fill one pan to two-thirds, or bake the extra batter separately as a few cupcakes.
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.