Before we discover how numerous pennies can fit in one square foot, we need to ask some questions...

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any minimum room required in between pennies?what pattern?are over there sides come the square foot and also no penny have the right to overlap any edge?other

Let"s start with a piece of cardboard that actions 12 by 12 inches, i beg your pardon is specifically 1 SF. Together you can see in the pictures, the ice cream measure mirrors we have 12 in. On both sides, even though it might be hard to check out the small numbers.



We start placing pennies top top the edge of this 12 in. Cardboard and it outcomes in a perfect fit: 16 pennies beside each other measure exactly 12 inches (one foot) since the diameter of a coin is .75 in. (or 3/4 of one inch).

By the way, as you can see, we supplied all brand new pennies because that this demonstration.

1 SF cardboard and 16 pennies top top its edge

Simple math have the right to tell us how many pennies to the right in one square foot in this right pattern (16x16=256) without in reality filling the square cardboard through pennies. We decided to perform it quiet in stimulate to show other details and for you to see how beautiful it looks. Here us go, adding row after heat of 16 pennies each. 

Worth mentioning right here is that the pennies execute touch every other. If a gap/space is forced all approximately each penny, say because that grout, transforms must be made.   

Slowly adding up pennies...

So, how numerous pennies deserve to fit in one square foot?256 pennies per square foot if the rows room straight

No coin overlaps any kind of edge and there"s no room left whatsoever - at least that"s what math tells us. If you see slight imperfections in the snapshot below is due to the fact that we inserted all 256 pennies by hand and also they are simply sitting there, no glued.

If a penny was square shaped instead of round, v a .75 in. Side, the cardboard below would be completely covered by 256 square pennies. But since the penny is round, you can see the cardboard in in between pennies.

Notice the empty area between any 4 pennies is quite huge and has 4 rounded sides (more ~ above this later).After closely placing every coin by hand, right here it is: 16 rows of 16 pennies each.

16 pennies per straight row x 16 rows = 256 pennies

Here"s a close-up ~ above a corner of the square cardboard. Also though the pennies space in call with every other, there"s still fairly some room in between them and also that"s as result of the right rows pattern/layout. 

Close-up on directly rows the pennies

And here"s another view that the exact same 256 brand brand-new shiny pennies sit in directly rows on a specifically one square foot cardboard.

256 brand new pennies on a 1 SF area
How many pennies have the right to fit in 1 sq. Ft. If we change the sample to staggered/offset rows?

The very first row of 16 pennies remains the same and we relocate the second row to the ideal by half a penny. Then we can also push up (as girlfriend look at the picture) the whole second row till it touches the first row the pennies.

We"ll press every also numbered row (2nd, 4th, 6th, ...) come the right by fifty percent a penny and then the whole row increase a little bit to touch the previous row. Through every row propelled up a little, us should get some room at the bottom of the cardboard.

Beginning of offset/staggered rows the pennies

Here"s more of a visual: advertise a heat by fifty percent a coin (A) results in half a penny overlapping the edge of ours square foot (B). 

Overlapping can"t happen if your an exact one square foot (SF) project has sides/walls like a tray. And also for enlarge projects, every SF of pennies have the right to overlap top top the following SF until you with the finish side of your project and also cannot fit another full coin anymore.

So, advertise 8 rows to the ideal by fifty percent a penny and then up a little, ours 16 initial rows the pennies got "squeezed upwards" closer together and revealed fairly the extra room (C) at the bottom of the cardboard.

Can we fit 2 much more rows that pennies there? We certain can. Not just that but there will certainly be a bit of room left after that which we"re no going come (mathematically) gain into, yet some much more "slices of pennies" will certainly fit in the tiny room left below. 

There we have actually the 2 extra rows that pennies (above) and also room left for "16 slices of pennies" aligned with the cardboard"s bottom edge.

How around the optimal side the the cardboard? one more 16 super small "slices that pennies" will fit in there, aligned v the edge, to make our square foot... Full.

And the overlapping pennies on the right side, compensate because that the north spaces top top the left, therefore no much more explanation needed here.

Easy math gives us the total variety of whole pennies (18x16=288) plus some 16 "slices of pennies" at the bottom the cardboard and also another 16 tiny slices in ~ the top.

The object of "How many whole pennies room in the 32 slices" can it is in the location of a brand-new article i m sorry is beyond the objective of this page. Yet if you"re a genius mathematician who wants to give it a shot, let united state know and we"ll publish your article and also give friend the credit.

A rapid eyeballing claims that the 32 slices can make up for about 6-8 pennies however let"s wait because that Einstein"s confirmation.

Here"s the humble conclusion...

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Pennies every square foot (sf) in balance out pattern:288 plus about 6-8 pennies "sliced" at optimal & bottom of cardboardThe complete could it is in 294-296 pennies every SF

What if her perfect square foot was a tray (or similar) with edges/sides i beg your pardon don"t enable pennies come overlap... Like the 9 pennies perform in the above or below picture.

So what then? Someone might say "let"s reduced the 9 overlapping pennies in half and fill the left side with the 9 halves". We entirely advise against cutting pennies. Simply, not incorporate the 9 overlapping pennies and the ideal side will be identical to the left.

Also, slide the whole thing down so the top and bottom will have actually identical spaces come the leaf of the square foot. Pennies per SF is 279 in this situation (288-9=279).