Coin density test.

Two ways to get the volume

Density is mass divided by volume, and the hard part is the volume of a small, irregular object. Water displacement reads it directly: the rise in water level, in millilitres, equals the coin’s volume in cubic centimetres. The hanging method uses Archimedes’ principle instead. Wikipedia’s hydrostatic weighing article gives relative density as W_air / (W_air − W_liquid), where W_liquid is the weight when submerged. Multiply by the water’s density to get density in g/cm³. Wikipedia’s water article gives 0.997 g/mL at 25 °C and 0.99997 at 4 °C, so the default here is 0.997 and you can change it.

Why the answer comes with a range

Each reading is rounded by the instrument. This tool takes half the scale step as the possible error on each weight, and one cylinder mark as the possible error on the volume (two readings, each half a mark off), then reports the lowest and highest density those errors allow. The worst case is deliberately pessimistic. A 31.10 g coin that raises the water from 20.0 to 23.0 mL on a cylinder marked every 0.2 mL gives 10.37 g/cm³, with a range of 9.72 to 11.11. The hanging method on the same coin with a 0.01 g scale gives 10.44 to 10.51, far tighter. For small coins the scale method usually wins.

Reference densities

The table in the result uses the recommended solid densities in Wikipedia’s “Densities of the elements (data page)”, near room temperature: gold 19.3, silver 10.49, copper 8.96, nickel 8.908, lead 11.34, palladium 12.023, tungsten 19.25, platinum 21.45 g/cm³ and so on. That page’s individual sources disagree on platinum (21.09 to 21.5), a reminder that the last digit is soft. These are pure elements. An alloy such as a silver and copper coin falls between its metals, so a 90% silver coin reads below 10.49.

Worked example

A coin of 31.10 g, the mass Wikipedia gives for the American Silver Eagle (31.103 g). If it is pure silver at 10.49 g/cm³, its volume should be 31.103 ÷ 10.49 = 2.965 cm³, so it should weigh about 31.10 − 2.965 × 0.997 = 28.14 g under water. Entering 31.10 and 28.14 returns 10.48 g/cm³, within 0.1% of silver and inside the error range. These readings are worked predictions for illustration, not a measurement of a real coin.

What density cannot tell you

Tungsten (19.25) and gold (19.3) differ by about 0.3%, far less than any home scale resolves, so density alone cannot separate them. The CoinTalk thread says the same in plainer words: if the result matches pure gold you may need a further test “because that density could be faked”. A mismatch does not prove a fake either: a worn, holed or alloyed coin has a different density.