Carats is a unit of mass used for diamonds, pearls and gemstone. One Carat is 0.2g or 0.00705oz, and his abbreviation is 'ct'So for this reason when be speak about the carats of the diamond we are speaking about his weight and not his size. If we take a few certificate diamonds of 1ct we will see that the measurements are different, and this is because it's a unit of weight and not a unit of volume.
Long time ago, the people had the belief that the Carob seeds had the exact same weight, so in 1570s was used to get the diamond's weight. At that time five carob seeds were 1ct.
As you can imagine, all the carob seeds doesn't have the same exact weight, this is why each country adopted a different weight unit standard for the diamonds weight.
The current standard (1ct = 0,2 g) was officially accepted in 1907 by General Conference of the Metric Convention.
We can use now a days very accurate scales, like the scale in the picture, that can get the weight very precisely. As you can see all the scale it's covered with glass to avoid any imprecision from the exterior.
As a conclusion, you have to know that carats (ct) is not the same as karat (k or kt). Karats with K it's a unit of purity used in precious metals as gold, silver or platinum.