Carats and millimeters do not correlate directly.
If, however, your question is about a gemstone cut as a round brilliant gem, it may weigh between .40 and .50 carats or 40/100ths or 50/100ths of a carat.
Take your gemstone to a jeweler whom you trust and ask that the diamond be removed from its setting and weighed. That is the only way to know the exact weight of any gemstone.
Both Asscher cut and Princess cut diamonds have at least one measurement that is 5mm, and both with this perimeter weigh in the one carat range. ('Square' diamond is a description for a rather generic cut.) A certified gemologist can give you an exact weight for your diamond.
If the diamond is measured in points -- 106, you could translate this to 1.06 carats.
If the diamond is a round brilliant cut, the 3mm measurement at the girdle indicates that the diamond weighs about 1/10th of a carat, or .10 carats.
If the diamond is a round cut, a stone weighing a little over 3 carats would have the 9.5mm measurement at the girdle.
The diamond weighs 112 carats, or 0.790136 ounces.
A one-half carat diamond, in a round brilliant cut, measures about 5.2MM at the girdle.
Both Asscher cut and Princess cut diamonds have at least one measurement that is 5mm, and both with this perimeter weigh in the one carat range. ('Square' diamond is a description for a rather generic cut.) A certified gemologist can give you an exact weight for your diamond.
There are 45.52 carats on the hope diamond
45.52 CaratsThe Hope Diamond is 45.52 carats. By comparison the average engagement ring is only 0.3 carats.
A round cut diamond weighing .25 carats measures 4.1mm at the girdle.
You're attempting to determine the weight of a stone given its measurements, which can not be accurate, since the two units of measurement are different. Roughly, however, if the stone is round, it may weigh between .40 and .50 carat, if the 5mm measurement is at the girdle of the diamond.
According to Wikipedia, below, this diamond weighs "755.5 carats (151 g)."
A round cut diamond weighing .25 carats measures 4.1mm at the girdle.
This diamond could weigh about six carats if its measurement is 10mm x 10mm.
If the diamond is measured in points -- 106, you could translate this to 1.06 carats.
From Wikipedia: "The Cullinan diamond is the largest rough gem-quality diamond ever found, at 3,106.75 carats (621.35 g)."
You can read more about the Spirit of de Grisogono black diamond, below, which weighs 312.24 carats.