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### Packing Pennies in a Jar

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Date: 06/08/99 at 09:55:41
From: Greg Gibson
Subject: Surface Area Question

What is the area of a penny? If a jar has a height of 11 inches and a
radius of 7 inches and it is full of pennies evenly to the top, how
many pennies can fit in this jar? I don't have anything with which to
measure the height of a penny, and I'm really not sure which formula
to use once I know the actual dimensions. Can you help?
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Date: 06/08/99 at 12:14:56
From: Doctor Peterson
Subject: Re: Surface Area Question

Hi, Greg.

What you're really interested in is not the surface area of a penny,
but its volume; and not really its volume, but the volume of the space
it will take up when it fits in a jar with other pennies, including
the wasted air space around it that no penny will fit into. If the
pennies are all lying flat so no space is wasted between layers, you
could approximate this by picturing each penny as a hexagon, to
include the "corner" space between pennies when they form an
approximately hexagonal array. The area of this hexagon would be six
times that of a triangle with height r (the radius of the penny) and
base 2r/sqrt(3), and the volume is that times the height h of the
penny:

V = 6 * 1/2 * r * 2r/sqrt(3) * h
= 2 r^2 h sqrt(3)
= 3.464 r^2 h

+--ooooooooo--+
/ooo         ooo\
/o  \         /  o\
oo    \       /    oo
o       \     /       o
/o        \   /        o\
/o          \ /          o\
+-o-----------X-----------o-+
\o          /|\          o/
\o        / | \        o/
o       /  |  \       o
oo    /   |r  \    oo
\o  /    |    \  o/
\ooo    |    ooo/
+--ooooooooo--+
2r/sqrt(3)

(By the way, this gives a crude approximation of pi.)

You'll have to find a way to measure the penny; I would suggest
stacking up enough of them to make several inches, and dividing that
distance by the number of pennies.

If you really want accuracy, you would have to find how many pennies
really fit into a given volume experimentally, to take account of how
they fit; you could do this by filling a known container and, if you
don't want to actually count, weighing them and dividing by the weight
of one penny.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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