MoneyBox/DensityAnalysis

Revision as of 10:44, 24 January 2004

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

20c

I've not caclulated this out, but from experience, $40 of 20c coins will fill a 1litre container nicely.

10c

The volume of a cylinder is determined as it's circular cross sectional area multiplied by it's height. Area is pi*(r^2). So, for a 10c coin, the volume is 1.98*pi*(11.8^2)=pi*275.7 which is roughly 866.12 cubic mm

So, it will take 1154 10c coins will occupy the volume of a litre. That's a theoretical maximum of course, and for a real-world litre container, there will be spaces between the coins.

So, let's assume coins stacked in piles, and those stacks arranged in a grid. ie, as if the coins were square rather than round. In this case, each coin takes up the volume of 23.6^2*1.98 = 1102.8 cubic mm.

With out 1million cubic mm litre, we can fit 906 coins in via this method.

In the real world, coins even dumped in randomly, will arrange themselves much more efficiently than this method, but the math to work it out would be quite horrible, and would need to know the container shape. If we worked out the volume according to piles arranged in a honeycomb, then that would probably get us much closer.

Hexagon stacking:

Same idea as above, but we work out the volume of a hexagon that is tangiential 6 times to the circle of the coin. We know the diameter of the coin, and from this the area of the hexagon in question is (sqrt(3)*d^2)/2

So for our 10c coins, this gives us a volume of 955.036 cubic mm - fitting 1047 coins into our million cubic millimetre area.

Conclusion: In the real world, you can probably get about 1000 10c coins into a 1litre container - or $100

5c

Volume per coin = 1.55*pi*9.705^2 = 145.99*pi = 458.638 cubic mm

This gives us 2180 coins to occupy 1litre of space.

Squared circle estimate gives us: 19.41^2 * 1.55 = 583.96 cubicmm

This gives us now only 1712 coins.

The hexed circle estimate: volume = ((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 lucky to squeeze 2000 5c coins into your 1litre container - or just on $100

Links

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