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Does Anyone Know How to Figure Out the Area of a Coin?
Coinosaurus
Posts: 9,644 ✭✭✭✭✭
Coinosaurus
Posts: 9,644 ✭✭✭✭✭
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Proud recipient of Y.S. Award on 07/26/08.
-Paul
SA = 2(pi * r^2) + (circumference * planchet width)
But if you want the more complex answer with all of the minted impressions (US of A, Liberty, etc.) then I don't want to think of how long that would take!
Do you think the raised areas would change the surface area much? Especially for modern coinage as flat as it is.
Regarding the volume of a coin, the best way to do it is to put in a container of water and see how much it displaces.
<< <i>Happy Pi Day! >>
I think we have a winner.
===========================
Total Surface Area = 2 * pi * r * (r + t)
r=radius
t=thickness
is an estimate.
the surface area would be more closely approximated with a laser scan of the surfaces and some numerical interpolation.
My friend, a former calculus teacher, figured out the volume of the keg, which was wider at the middle than it was at the ends. We then figured out the volume of a large cent. He rough-guessed that you'd have to allow 20% for the fact that the coins are solid with gaps in between, rather than liquid. And what did we come up with?
About 14,000 large cents. Non-coincidentally, a newspaper ad I have from 1821 that offers cents from the US Mint offers them "by the keg, in quantities of $100 to $140." In other words, 10,000 to 14,000 coins.
And how did I know the keg was a coin keg and not a nail keg of similar vintage? It bore an old wax seal from a bank. I've been in my bank's vault -- lots of coins, not too many nails.
The person to whom I sold it donated it to the American Numismatic Society.
Long story short: study your geometry, kids. You never know when it'll help you sell a ratty old ugly wooden barrel.
Betts medals, colonial coins, US Mint medals, foreign coins found in early America, and other numismatic Americana
Add in the surface area of the rim. This is essentially a rectangle whose sides are the rim height (h) and coin circumference (pi*2*r). Multiply them.
Surface Area = 2(pi*r^2) + h(pi*2*r)
Lance.
reeds (and number of), incuse letters, smoth....
<< <i>pi r squared [times two for obverse and reverse] plus thickness times pi r for the rim area >>
Actually it is 2pi r or pi d x thickness for the rim.
<< <i>pi r squared >>
wrong. pie are round, cornbread are square.
RMR: 'Wer, wenn ich schriee, hörte mich denn aus der Engel Ordnungen?'
CJ: 'No one!' [Ain't no angels in the coin biz]
<< <i>pi r squared >>
Pi r round
<< <i>
<< <i>pi r squared >>
wrong. pie are round, cornbread are square.
OK... that made me laugh...
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proof bag- fill it with water and weigh it - 1ml( or 1 cubic centimeter ) of water is pretty close to
1 gram - you'd have tare the keg and bag first of course
Generally in the vicinity of my front left pants pocket. Eventually, most wind up inside my son's piggy bank.
(With apologies to RWB...
<< <i>got any pics of the coin keg and its seal? >>
Ask and ye shall receive.
Betts medals, colonial coins, US Mint medals, foreign coins found in early America, and other numismatic Americana
Worry is the interest you pay on a debt you may not owe.
"Paper money eventually returns to its intrinsic value---zero."----Voltaire
"Everything you say should be true, but not everything true should be said."----Voltaire
Same with coins. If the design is based on a master die or hub, calculating the surface area including the design and reeding will be theoretical, and not exact. With the new digital modeling software used for coin designs, the exact surface area can be determined.
That is, if anyone is crazy enough to want to know the exact surface area of a coin
<< <i>I understand we are now supposed to use "tau" instead of "pi." "Tau" is 2 * pi.
Regarding the volume of a coin, the best way to do it is to put in a container of water and see how much it displaces. >>
This is the most accurate answer.
<< <i>pi r squared [times two for obverse and reverse] plus thickness times pi r for the rim area >>
That's an underestimate. Struck coins aren't a perfect cylinder. Close, but no cigar.
I do not believe it is possible to calculate an accurate surface area of a coin through simple mathematics. The designs are too complex (because of the reeding & design elements in relief). I would imagine one could experimentally calculate it within some error rate, but I am not familiar with the technique used on something so small....
I know when I was faced with the a similar problem as an engineer many years ago (the surface area of an airplane part), I took paper and covered the complex curved surface, then measured the resulting paper when flat. It was close enough for the task at hand (which was essentially how much paint we needed to cover X parts).
All of the above said, because of the nature of a coin's surface, the closer you look, the more surface you see, so the true surface area at a microscopic level is much, much larger than at the macro level. Therefore, calculating surface area is always a bit of a loaded question when you're dealing with real elements (like metals) and not perfect theoretical materials.
Just some food for thought, respectfully submitted.
<< <i>
<< <i>I understand we are now supposed to use "tau" instead of "pi." "Tau" is 2 * pi.
Regarding the volume of a coin, the best way to do it is to put in a container of water and see how much it displaces. >>
This is the most accurate answer. >>
To a question not asked.
The OP is interested in surface area, not volume, I do believe.
<< <i>
<< <i>got any pics of the coin keg and its seal? >>
Ask and ye shall receive.
Reminds me of the old song "Keg of my heart......"
On a sub-atomic scale, surface area is undeterminable. Compared to the diameter of an individual atom, there are vast spaces between molecules, and certainly vast spaces between the nucleus of said atom and the outmost reaches of its orbiting electrons.
Surface area can only be determined upon theoretical planes and mathematically defined objects. So I guess those of you who are intent on defining an exact answer are on a fool's errand.
This reminds me of a joke told in physics classes:
<< <i>Three scientists are each asked to determine how many eggs a chicken will lay each month.
The biologist employs knowledge of the chicken's size, age, weight, etc...
The mathemetician employs knowledge of statistical analysis...
The physicist shakes his head and states, "No! You must first assume the chicken is a sphere..." >>
Actually, I only heard this told once, in my honors physics class in high school. A few students laughed out loud, several groaned, but most were left waiting for the punch line.
Empty Nest Collection
<< <i>This reminds me of a joke told in physics classes:
<< <i>Three scientists are each asked to determine how many eggs a chicken will lay each month.
The biologist employs knowledge of the chicken's size, age, weight, etc...
The mathemetician employs knowledge of statistical analysis...
The physicist shakes his head and states, "No! You must first assume the chicken is a sphere..." >>
>>
do chickens like pi?
<< <i>I do not believe it is possible to calculate an accurate surface area of a coin through simple mathematics. The designs are too complex (because of the reeding & design elements in relief). I would imagine one could experimentally calculate it within some error rate, but I am not familiar with the technique used on something so small.... >>
<< <i>when I was faced with the a similar problem as an engineer many years ago (the surface area of an airplane part), I took paper and covered the complex curved surface, then measured the resulting paper when flat. It was close enough for the task at hand (which was essentially how much paint we needed to cover X parts). >>
With CATIA V5 software, point and click will get exact surface area, I don't know if coin design software has the same ability. Without the help of a computer, I would agree it would be impossible to get a correct answer, but an area very close to the true surface area could be calculated. A struck coin would vary in surface area slightly, although the digital data of a computer design would have the elements needed for the correct surface area of the coin design.
Surface area from flat pattern is close enough for most purposes, for aerodynamics and tooling, digital data is much better. Surface area and volume can vary quite a bit depending on shape, the coins in the barrel will get volume but not surface area.
After reading Mr. Coinosaurus's book on the US Mint, I would expect that he wants nothing less than an exact answer to his question
I think we all know how to work with perfect circles, cylinders, and spheres. At least I hope so.
Regarding volume - I had a similar problem to Pistareen's. There was an 1870 Carson City newspaper article describing transport of CC silver dollars in a half-bushel container. After some analysis, it was clear that the container was nowhere near full. I did a test using only nickels, only dimes, and only quarters, to see how much volume they would displace in a container of water. The smaller coins "packed" better, ie. consumed more of the total available volume. As I recall the "packing factor" was in the 70%-80% range, similar to what Pistareen has reported. I also measured before and after "settling" (ie. shaking the container to get the coins to pack more densely) and this resulted in improving the packing by several percent. I asked Dr. John McCloskey, well-known numismatic researcher and PhD in statistics, if there was a more elegant way of making this calculation. He did not think there was an easy way to do it, although I think you could possibly use some kind of 3-D simulation software that could stack coins randomly.
Regarding surface area - this is also an interesting problem, again best left to a computer that can model 3-D objects. A crude way to measure it would be to dip a number of plain and struck planchets in adhesive & then compare the amount of adhesive remaining in the original container.
212 is New York City.
213 is Lost Angeles.
312 is Chicago,
313 is Detroit................
Sometimes, you brothers employ too much deep thinkin' for an old man like me.
<< <i>
<< <i>I do not believe it is possible to calculate an accurate surface area of a coin through simple mathematics. The designs are too complex (because of the reeding & design elements in relief). I would imagine one could experimentally calculate it within some error rate, but I am not familiar with the technique used on something so small.... >>
<< <i>when I was faced with the a similar problem as an engineer many years ago (the surface area of an airplane part), I took paper and covered the complex curved surface, then measured the resulting paper when flat. It was close enough for the task at hand (which was essentially how much paint we needed to cover X parts). >>
With CATIA V5 software, point and click will get exact surface area, I don't know if coin design software has the same ability. Without the help of a computer, I would agree it would be impossible to get a correct answer, but an area very close to the true surface area could be calculated. A struck coin would vary in surface area slightly, although the digital data of a computer design would have the elements needed for the correct surface area of the coin design. >>
It was a complex curved part/surface I could not reproduce in 3d on AutoCAD 20 years ago.
Your point is a good one for the coin problem, but I might solve it a different way....as I understand laser scanning can reproduce even complex solids on the computer (and presumably import them into CATIA or an equivalent), then as you point out, it would take a click of the button to figure out a very close approximation to surface area.
Very interesting question and discussion (and sorry, Curly
<< <i>An easy way to measure the volume of the keg- put in a water
proof bag- fill it with water and weigh it - 1ml( or 1 cubic centimeter ) of water is pretty close to
1 gram - you'd have tare the keg and bag first of course >>
But if you tare the keg and bag, the water (and then the large cents) will fall out!
An authorized PCGS dealer, and a contributor to the Red Book.
<< <i>Your point is a good one for the coin problem, but I might solve it a different way....as I understand laser scanning can reproduce even complex solids on the computer (and presumably import them into CATIA or an equivalent), then as you point out, it would take a click of the button to figure out a very close approximation to surface area.
Very interesting question and discussion (and sorry, Curly
This makes me wonder if the PCGS sniffer is setup to calculate surface area, and if so, can this be used as a "fingerprint" for the coin (aside from the complete 3-D representation in software, of course). I suspect this feature is included in their software, although they may not make use of it.
<< <i>Your point is a good one for the coin problem, but I might solve it a different way....as I understand laser scanning can reproduce even complex solids on the computer (and presumably import them into CATIA or an equivalent), then as you point out, it would take a click of the button to figure out a very close approximation to surface area. >>
I also believe the technology for 3D laser scanning exists to re-create the surface model of a coin. Could be used for counterfeiting detection.
<< <i>
<< <i>Your point is a good one for the coin problem, but I might solve it a different way....as I understand laser scanning can reproduce even complex solids on the computer (and presumably import them into CATIA or an equivalent), then as you point out, it would take a click of the button to figure out a very close approximation to surface area. >>
I also believe the technology for 3D laser scanning exists to re-create the surface model of a coin. Could be used for counterfeiting detection. >>
,,,,,,,or for creating counterfeit dies.
Worry is the interest you pay on a debt you may not owe.
"Paper money eventually returns to its intrinsic value---zero."----Voltaire
"Everything you say should be true, but not everything true should be said."----Voltaire
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