Area scaling versus volume scaling
Area scaling is the right choice when the batter depth will stay roughly the same. That covers many swaps between standard round tins, square tins, and sheet-style cake pans where the depth is similar.
Volume scaling becomes more useful when the pan depth changes materially as well. In that case, comparing only the footprint can miss how much extra batter the taller pan can hold or how much thinner the batter will sit in a shallow pan.
Round area = pi x (Diameter / 2)^2
Used for round cake tins and springform-style pans.
Rectangular area = Width x Length
Used for rectangular trays, sheet pans, and loaf-style pans.
Scaling factor = Target area or volume / Original area or volume
The multiplier applied to the original batter quantity or to any example ingredient amount.