A geometric formula (or other constant ratio) must form a constant ratio to be consistent (and useful). Thus, the "ratio to create the rule" must be one that sets the desire ratio to some constant.
This says that whatever the values of the volume and the radius are, this ratio will result in a constant number. Proving it to be correct would require tests on many different actual values.
One common example is the definition of the value
From a common equation we can see its derivation as the ratio (as in this problem) of the circumference of a circle to its radius: