Area and Volume of a Football

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 
would one find the volume of a football?" Please help 
me with these problems if you can. Thanks.

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.

        I hope this answers your question; if you have more 
questions or anything is unclear (this is hard stuff) please 
feel free to write back.

-Dan "Dr. Math" Eisenbud