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Anne Marie Helmenstine, Ph.D.

How to Measure Volume & Density - A Tale of Archimedes

By August 5, 2007

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Archimedes needed to determine if a goldsmith had embezzled gold during the manufacture of the royal crown for King Hiero I of Syracuse. How would you find out if a crown was made of gold or a cheaper alloy? How would you know if the crown was a base metal with a golden exterior? Gold is a very heavy metal (even heavier than lead, though lead has a higher atomic weight), so one way to test the crown would be to determine its density (mass per unit volume). Archimedes could use scales to find the mass of the crown, but how would he find the volume? Melting the crown down to cast it into a cube or sphere would make for an easy calculation and an angry king. After pondering the problem, it occurred to Archimedes that he could calculate volume based on how much water the crown displaced. Technically, he didn't even need to weigh the crown, if he had access to the royal treasury, since he could just compare the displacement of water by the crown with displacement of water by an equal volume of the gold the smith was given to use. According to the story, once Archimedes hit upon the solution to his problem, he burst outside, naked, and ran through the streets yelling, "Eureka! Eureka!"

I don't know if the part about Archimedes running naked through town is true, nor do I know how things turned out regarding the crown and the goldsmith. What I do know is you can use Archimedes' idea to calculate the volume of an object, and its density, if you know the object's weight. For a small object, in the lab, the easiest way to do this is to partly fill a graduated cylinder large enough to contain the object with water (or some liquid in which the object won't dissolve). Record the volume of water. Add the object, being careful to eliminate air bubbles. Record the new volume. The volume of the object is the initial volume in the cylinder subtracted from the final volume. If you have the object's mass, its density is the mass divided by its volume.

Most people don't keep graduated cylinders in their homes. The closest thing to it would be a liquid measuring cup, which will accomplish the same task, but with a lot less accuracy. There is another way to calculate volume using Archimede's displacement method. Partially fill a box or cylindical container with liquid. Mark the initial liquid level on the outside of the container with a marker. Add the object. Mark the new liquid level. Measure the distance between the original and final liquid levels. If the container was rectangular or square, the volume of the object is the inside width of the container multiplied by the inside length of the container (both numbers are the same in a cube), multiplied by the distance the liquid was displaced (length x width x height = volume). For a cylinder, measure the diameter of the circle inside the container. The radius of the cylinder is 1/2 the diameter. The volume of your object is pi (3.14) multiplied by the square of the radius multiplied by the difference in liquid levels (pr2h).
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Comments

August 16, 2007 at 6:45 am
(1) Robert Love says:

You say:
“Technically, he didn’t even need to weigh the crown, if he had access to the royal treasury, since he could just compare the displacement of water by the crown with displacement of water by an equal volume of the gold the smith was given to use.”
Archimedes still needs the mass (or weight at the same place on Earth) of the gold. — If the goldsmith replaced pilfered gold with an equal volume of lead, for instance, and mixed it well with the remaining gold, there is no way to tell feom the volume displaced without the mass as well. The DENSITY, requiring both mass AND volume, of the crown gives the answer, not the volume alone.

August 21, 2007 at 11:44 am
(2) Jay says:

Dactual = Mair/(Mair-Mwater) Or simply stated the density of an object is equivalent to the mass in air divided by the difference between its mass in air and its mass in water.

November 6, 2007 at 4:49 pm
(3) madbelgian says:

“Technically, he didn’t even need to weigh the crown, if he had access to the royal treasury, since he could just compare the displacement of water by the crown with displacement of water by an equal volume of the gold the smith was given to use.”

You are correct in saying that it is not necessary to know the weight of the crown to know if this is gold, however, the way it is described is simply wrong.

If you replace a metal by another metal occupying the same volume, they will both displace the same volume of water. On the other hand, if you can compare the displacement of water of the crown to the displacement of water of the same weight of gold (knowing the exact weight is irrelevant to the problem) from the treasury. If the displacements are equal the crown is made of gold.

One could imagine having a mixture of metals with an average density equal to the density of gold. Only tungsten’s density is very close to gold’s density, and platinum’s is higher. If it is platinum don’t say anything and wear it proudly :)

September 11, 2009 at 10:59 am
(4) loser says:

this didn’t tell me how to measure my body fatat all you idiot thanks to you i still eat twinkies for breakfast1 $#@$%@$

September 15, 2010 at 8:38 pm
(5) Barnes Powell says:

Actualy the King got another Crown and the goldsmith head was chopped off for trying to trick the king.

October 7, 2010 at 9:51 am
(6) lol brown says:

lmfao!!!!!!!!

August 29, 2011 at 9:57 pm
(7) User101 says:

How did he do it????

November 6, 2011 at 8:51 am
(8) Emily says:

i think this is helpful, though some parts are difficult to work out what is meant. thank-you! :)

October 4, 2012 at 8:30 am
(9) anonymous says:

he actually said Eureka for some thing else!!!!!!!!!!!!!!!

October 29, 2012 at 7:54 am
(10) anoop says:

hi mam,

Please tell how do i check a hollow gold balled ornament using density test. Or any other way to identify the balled ornament is gold ? please reply me

December 11, 2013 at 2:24 am
(11) Samuel Sikaona says:

How did he do it because its confusing Me….

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