Early Greeks could not measure volume to the same accuracy as can be done today, so this method may not have worked for Archimedes.A more practical approach is as follows:

First, Archimedes would confirm that the crown had the mass of the gold provided originally: 3014 g.

Next, he tied the crown to a string tied to one end of the scale. Then he submerged the crown under water and found the mass required to exactly balance the scale. Let's say the scale balanced at 2845.4 g.

The difference between the mass of the crown in air and its mass while submerged is the mass of the water displaced by the crown.

The mass of the water provides a way to calculate the volume of water, provided you know that the density of water is

\fCalculation Result Unit Mass of crown 3014 Mass that balances the crown when it is under water 2845.4 g Mass of water displaced =Mass of Gold - Mass of Gold under water Density of water 1.00 g/ cm3 Volume of displaced water with this mass = Mass water displaced Density of water So the volume of the crown is Now use mass and volume of the crown to calculate its density D = . = Is this pure gold