Why is circumference important?

Say you inherit a set of glass mugs. If one of them breaks, and you want a mug that holds the same exact amount of liquid or whatever else you'd put in it, you want to be able to compare the circumference of your mug to the circumference of any other same sized cup. It sounds kind of stupid because I doubt anyone would care about finding the perfect mug, but that's an example. It's important if you want to be a math teacher or scientist in the future. Lets just say that.