How many nickels would fit in a square foot?

Answer 1

Great question !

First of all, let's assume that the square foot is in fact a foot square (i. e. 12" x 12", rather than, say, 6" by 24"). According to the related link, a U.S. nickel is 0.8125 inches in diameter. 12" / 0.8125" = approx. 14.77. So, you can put 14 in a row along an edge, with a bit of room left over.

Now place the next row so that the coins fit in the gaps of the row below. If this pattern is continued, you get something that looks a bit like a honeycomb. With a bit of trigonometry (the 30-60-90 triangle), we can show that each row will take only rad(3)/2 = 1.732/2 = 0.866 times the diameter of the nickel.

This would be 0.866*0.8125" = approx. 0.7036". Now, take 0.8125" from 12" for the first row: 12-0.8125 = 11.175. Divide 11.175/0.7036 = approx 15.88, so you can get 1+15 = 16 rows.

16*14 = 224 nickels in a square foot.

But that may not be the final answer ! The 0.88 rows left over suggests that if you shuffle things around a bit, you may be able to fit in one or two more. In fact, this is called a "packing problem", and there is no known formula for predicting the exact number of circles that will fit within a given shape.