Filled to the rim and filled to the brim both make sense. Find out which phrase is more common.
On Twitter, @PlainTeaGirl asked whether a cup is full to the rim or full to the brim.
Rim and brim both refer to the edge of a glass or cup, but there is one small difference: The entry at Dictionary.com notes that whereas rim refers generally to the edge of a cup or glass, brim usually refers to the inside edge. Further, in the late 1600s, people started using brimming as an adjective to describe streams and rivers that were filled to overflowing. According to the Oxford English Dictionary, brimming was first used in writing by John Milton in Paradise Lost.
Rim, on the other hand, has not spawned an adjective form that means overflowing.
The American Heritage Dictionary of Idioms says that filled to the brim is the correct phrase, and a Google Ngram search, which shows how often phrases are used in books in the Google Books collection, also shows that filled to the brim appears much more often than filled to the rim. The graph is similar when you compare full to the rim and full to the brim.
Verdict: Stick with filled to the brim.