Converting a 9″ square 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
A 9″ square recipe comes to about 1⅔ 8″ round pans, which doesn't divide neatly. Three honest options: fill two pans and accept that the last comes out shallower; scale the recipe by 0.60× for a single pan; or scale it by 1.20× to fill both properly. 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.