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### Area and Volume of a Football

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Date: 28 Mar 1995 12:08:04 -0500
From: Mary Basse
Subject: geometry question

Hi. My name is Russell Heinrichs. I am a freshman at
Anna-Jonesboro High School. A couple days ago when
I was pumping up my football, I thought, "How would
one find the area of a football? Or then again, how
me with these problems if you can. Thanks.
```

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Date: 30 Mar 1995 00:42:39 -0500
From: Daniel Eisenbud
Subject: Re: geometry question

One way to physically find the volume of a football
would be to put some water in a container with straight
sides (a fish tank, or a big pot, or something) and
measure how much the water rises when you submerse
the football in it.  Then, if you can find the area of the
cross-section of the container of water, and multiply
that by the increase in height, you'll have the volume of
the football.
I can't think of any really easy way to physically
find the surface area of the football.  One way to approach
this problem might be to divide the football into one-inch
slices from end to end, and measure the how wide the
football is from edge to edge (perpendicular to the axis of
the football) at some point in each of these slices (for
practical purposes, the middle of the slice would be a good
place.)  Multiply the width of each slice by pi, and you have
the circumference of a circle in that slice.  Now multiply
that by the thickness (in this case 1") of the slice, and
you'll get an approximation to the surface area.  Now add
up the approximations to the surface areas of all the slices,
and you'll have something close to the surface area of the
football.  A theorem in integral calculus says that as the
width of the slices approaches zero, and therefore the
number of slices approaches infinity, the sum will approach
the actual surface area of the football.  So you would get a
better approximation if you measured every quarter-inch
than if you measured every inch.  You could do a similar
thing to mathematically approximate the volume: find the
area of a cross section of each slice, which would be
pi*d^2/4 (which is equivalent to pi*r^2), multiply it by the
thickness of the slice, for an approximation of the volume
of the slice, and add them all up.

questions or anything is unclear (this is hard stuff) please
feel free to write back.

-Dan "Dr. Math" Eisenbud
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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