Here, let me be the first to suck the humor out of this:

I think the price comes out to more like $2.97 per square foot. Anyone care to either confirm or refute that? If I recall, a penny is 19mm in diameter, and they appear to be packing them like hexagons.

I'm trying to come up with a comment about the labor cost that is both socially relevant and a clever play on words, but I got nothing.

Regardless, how does the math work to get to $2.97 per square feet? It obviously isn't $1.44, which I am thinking was calculated assuming pennies that are one-inch diameter, in a square pattern (which clearly, they are not).

Apothem is the distance from the center point to the center of the edge.

I calculate the apothem at 9.525.

That makes a side about 11mm, so the area of one penny - if it were a pefect hexagon - at 314.325 mm^2.

Google says 1 square foot = 92,903.04 square mm.

That means 295.5ish pennies.

Factor in a few rounding differences between Ryan and me, and I'd say we agree.

Ryan McGuire
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Henry Wong wrote:
Regardless, how does the math work to get to $2.97 per square feet? It obviously isn't $1.44, which I am thinking was calculated assuming pennies that are one-inch diameter, in a square pattern (which clearly, they are not).
Henry

If the pennies are packed in a hexagon pattern (or do you call that a triangular pattern), we can pretend that they are hexagons with a side-to-side (as opposed to corner-to-corner) diameter of 19.05mm or 0.75in. The area that hexagon is 0.75^2 * sqrt(3) / 2 or 0.48713928962 in^2 *. How many of those would fit in 144 in^2? 144/..487 = 295.6something. 295 pennies would cost approximately $2.95.

* As luck would have it, I remember from high school the formula for the area of a hexagon given its side-to-side diameter, but I still have to work it out when give the length of one side. A = D^2 * sqrt(3)/2

Oh, I get it !! I was wondering why hexagon approximations were used. You are not concerned with the area of the penny, you are concerned with the amount of coverage it was doing. And that means that it needs to calculate the space between the pennies that were not being covered.

Great job. I am giving cows all-around !!

Ryan McGuire wrote:
* As luck would have it, I remember from high school the formula for the area of a hexagon given its side-to-side diameter, but I still have to work it out when give the length of one side. A = D^2 * sqrt(3)/2

BTW, I never knew the formula for a hexagon. I had to calculate it as six equilateral triangles, where the height is the radius of the penny, and the base is calculated via trigonometry (of a 30/60/90 triangle).

Henry Wong wrote:BTW, I never knew the formula for a hexagon. I had to calculate it as six equilateral triangles, where the height is the radius of the penny, and the base is calculated via trigonometry (of a 30/60/90 triangle).

I had to look up the formula, and the page I found did exactly what you describe to derive it.

If it were per square foot, then how does it matter if it is arranged as a hexagon? I admit that geometry was not my cup of tea ever, but still.
So I calculated using the total mm in a square foot given by fred which is 92,903.04 mm and dividing it by area of the penny which is 285.022 square mm.
Which gives me $3.26. Am I missing something obvious here??

Amit Ghorpade wrote:If it were per square foot, then how does it matter if it is arranged as a hexagon? I admit that geometry was not my cup of tea ever, but still.
So I calculated using the total mm in a square foot given by fred which is 92,903.04 mm and dividing it by area of the penny which is 285.022 square mm.
Which gives me $3.26. Am I missing something obvious here??

Yes. You are forgetting about the gaps between the pennies. The reason that the pennies are treated as a hexagon, and not as a circle, is because that is the shape that it is taking up on the floor (from the picture). In your case, since you are not allowing for that, the extra 28 or 29 cents is needed to get enough metal, which presumably needs to be melted down, to fill in those gaps.

A penny is a circle. you cannot get solid coverage with circles, as there are spaces between them.

You can with hexagons, or squares, or many other shapes...

Take a look at this. You can see some shpaes allow complete coverage with no overlaps or gaps, and others don't. There are also better ways to 'pack' them in. If you aligned them in a square grid, you'd have larger gaps than if you off-set each row - you can kind of squeeze down the size of the gaps. I believe that packing them this way minimizes the gaps, and is best represented by a hexagon.

Ryan McGuire
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fred rosenberger wrote:in other words...

A penny is a circle. you cannot get solid coverage with circles, as there are spaces between them.

You can with hexagons, or squares, or many other shapes...

Take a look at this. You can see some shpaes allow complete coverage with no overlaps or gaps, and others don't. There are also better ways to 'pack' them in. If you aligned them in a square grid, you'd have larger gaps than if you off-set each row - you can kind of squeeze down the size of the gaps. I believe that packing them this way minimizes the gaps, and is best represented by a hexagon.

If you did pack pennies (0.75 in diameter) in a square grid, the math works out nicely. You'd end up with a 16x16 grid of pennies worth $2.56 in each square foot. Square grid -> larger gaps -> fewer pennies -> cheaper per square foot.

You guys are totally forgetting that pennies have two sides, so you have to divide by two; you use both sides of the pennies. Look carefully at the photo and you'll see both heads and tails. $2.95/2 = $1.44 (within error bars.) Sheesh.