Is quarter round molding a type of shoe molding, or are they different things?
Quarter round molding
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They're used almost interchangeably today, but no there is a slight difference.
Shoe molding is wider/taller (depending on orientation) in one direction than the other and the curve isn't a perfect arc, rather than being a perfect 1/4 of a circle as in quarter round.
I feel like this is one of those answers where more than one answer can be right. It boils down to experience and what you were taught.
I understand shoe molding as the molding family that is partnered with a baseboard. In the same way that crown molding describes the family of molding used to finish top edges.
When used on baseboard the quarter round is being used as shoe molding. It is still quarter round. I haven't found a reliable source worth linking too but seen mentioning that shoe molding is typically taller than wider for aesthetics reasons which gives credence to TX Turner's answer.