MoneyBox/DensityAnalysis

Revision as of 14:05, 15 January 2008

...a Density Analysis of australian coinage ie, how much value of a given coin can you fit inside, say, 1litre of space?

So what are we dealing with:

• 1litre = 1000cm^3 of volume = 1,000,000 cubic mm

Australian coin sizes: (see: http://www.worldmints.com/ccoin_ram.asp) (note that diameter is the specified, but thickness is the maximum legal allowed)

5c

diameter: 19.41mm, thickness: 1.55mm

10c

23.6mm, 1.98mm

20c

28.52mm, 2.52mm

50c

31.51mm (across flats), 2.8mm

$1

25mm, 2.8mm

$2

20.5mm, 3mm

How to calculate

There are three estimates easily possible.

1. Count of coins for maximum displacement within our 1litre limit (1litre divided by volume per coin)
2. Same as before, but assume coins are square with edge length equal to diameter. Simulates stacks arranged in a grid
3. Same as before, but assume coins are hexagonal and stacked in a hex lattice. (is this the most efficient possible?)

In the real world, coins are unlikely to be stacked neatly, but with repeated shaking should form themselves into dense stacks. Thus I'd expect a real-world result somewhere between the 'squared' and 'hexed' results.

Volume of coin

(pi*r^2) * thickness

Volume of squared coin

d^2 * thickness

Volume of hexed coin

((sqrt(3)*d^2)/2) * thickness

5c

real coins

• 1.55*pi*9.705^2 = 145.99*pi = 458.638 cubic mm
• This gives us 2180 coins to occupy 1litre of space.

Squared coins

• 19.41^2 * 1.55 = 583.96 cubicmm
• This gives us now only 1712 coins.

Hexed coins

• ((sqrt(3)*19.42^2)/2)*1.55 = 506.245 cubic mm
• This now gives us 1975 coins.

Conclusion

In the real world, you might get around 1800 5c coins into your 1litre container - or approx $90

10c

real coins

• 1.98*pi*(11.8^2)= pi*275.7 = 866.12 cubic mm
• This gives us 1154 coins to occupy 1litre of space.

Squared coins

• 23.6^2*1.98 = 1102.8 cubic mm.
• This now gives us only 906 coins

Hexed coins

• ((sqrt(3)*23.6^2)/2)*1.98 = 955.036 cubic mm
• This gives us 1047 coins into our million cubic millimetre volume.

Conclusion

In the real world, you can probably get about 950 10c coins into a 1litre container - or $95.

In an actual test however, 750 coins ($75) left little room available, and as coins become available, I doubt I'll get more than 800 in there.

20c

real coins

• 2.52*pi*(14.26^2)= pi*512.436 = 1609.86 cubic mm
• This gives us 621 coins to occupy 1litre of space.

Squared coins

• 28.52^2*2.52 = 2049.744 cubic mm.
• This now gives us only 487 coins

Hexed coins

• ((sqrt(3)*28.52^2)/2)*2.52 = 1775.13 cubic mm
• This gives us 563 coins into our million cubic millimetre volume.

Conclusion

In the real world, you might get around 500 20c coins into your 1litre container - or $100

$1

real coins

• 2.8*pi*(12.5^2)= pi*437.5 = 1374.45 cubic mm
• This gives us 727 coins to occupy 1litre of space.

Squared coins

• 25^2*2.8 = 1750 cubic mm.
• This now gives us only 571 coins

Hexed coins

• ((sqrt(3)*25^2)/2)*2.8 = 1515.54 cubic mm
• This gives us 659 coins into our million cubic millimetre volume.

Conclusion

In the real world, you're gonna be looking at around 600 coins. I'm not gonna even pretend I need to work out the dollar value here.

$2

real coins

• 3*pi*(10.25^2)= pi*315.19 = 990.19 cubic mm
• This gives us 1009 coins to occupy 1litre of space.

Squared coins

• 20.5^2*3 = 1260.75 cubic mm.
• This now gives us only 793 coins

Hexed coins

• ((sqrt(3)*20.5^2)/2)*3 = 1091.84 cubic mm
• This gives us 915 coins into our million cubic millimetre volume.

Conclusion

• You're probably looking at the 800-850 coins mark if you're lucky, or $1600+ :)

Links

• http://www.drking.worldonline.co.uk/hexagons/misc/area.html