How do you adjust a recipe for a different pan size?

For pans of similar depth, scale by surface area rather than by diameter or by eye. A round pan's area is pi times the radius squared, so a 9-inch round is about 64 square inches and an 8-inch round about 50. Moving a recipe from a 9-inch to an 8-inch means multiplying by roughly 0.8, or accepting a thicker cake with a longer bake. Rectangular pans are simply length times width, so a 9-by-13 is 117 square inches, which is very close to two 8-inch rounds at 100 or two 9-inch rounds at 127.

Bake time changes even when the volume is scaled correctly, because depth changes how heat reaches the center. A thicker batter needs lower temperature and longer time; a thinner one bakes faster and can overbrown at the edges before the center sets. The practical approach is to start checking at roughly two-thirds of the original stated time and judge by the recipe's doneness indicator, a clean tester, internal temperature, or the sides pulling away, rather than by the clock. Reducing oven temperature by about 25 degrees helps when going noticeably deeper.

Two cautions specific to baking. Chemical leavening does not scale perfectly, so very large adjustments in either direction can produce texture problems, and scaling beyond roughly double or half is where recipes tend to break down. And some formats do not convert well regardless of arithmetic: a recipe developed for a tube or bundt pan relies on the center column for heat penetration, and cheesecakes, angel food, and soufflés depend on specific pan geometry and surface behavior. When in doubt, filling any pan no more than two-thirds full is the rule that prevents overflow, and the excess batter can go into muffin tins, which bake in a fraction of the time.