Converting a 9″ round recipe to an 8″ square pan
Batter scales with volume, not with width. A 9″ round pan holds about 8 cups filled to the brim, and an 8″ square pan holds 8 cups — near enough the same that you can bake the recipe as written.
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″ round recipe fills about one 8″ square at the usual two-thirds level — a clean split, which makes this an easy conversion to do without scaling anything.
Cross-check against Wilton's chart: Wilton puts 5 cups of batter in one 2-inch layer of a 9″ round pan and 6 in an 8″ square pan, a ratio of 0.8 — against 1 by brim capacity. The two differ because Wilton fills each pan to its own level (anywhere from half to two-thirds); this converter uses brim capacity throughout so every pair is on the same footing.
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.