Converting a 9″ square pan recipe to an 8″ round pan
Batter scales with volume, not with width. A 9″ square pan holds about 10 cups filled to the brim, and an 8″ round pan holds 6 cups, so you need 40% less batter. Multiply every ingredient — including the eggs, which is the awkward one — by 0.60×.
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 1.1″. 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
One 9″ square recipe comes to roughly 1½ 8″ round pans, which doesn't divide neatly — you'd need 2 of them, with the last only part full. Two honest options: scale the recipe by 0.60× and bake a single pan, or make the full batch and accept that one comes out shallower than the rest. Batter left standing while an earlier pan bakes loses its lift, so get them all in the oven together where you can.
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.