The Slice Puzzle
Solution

Surprisingly, any pair of 1 inch slices of the earth will have the same surface area. Any pair of slices of the earth of the same thickness will have the same surface area. And a similar result is true for pairs of slices of the same thickness from any sphere.

The proof requires a computation of the surface area of a surface of revolution. For our case, revolve the curve

y = sqrt ( R2 - x2 )
about the x axis to get a sphere. The formula for the surface area of such a surface of revolution is
Surface area = Integral ( A <= X <= B ) 2 * pi * y(x) * sqrt ( 1 + (y')2 ) dx
The integrand can be simplified to:
2 * pi * sqrt ( R2 - x2 ) * sqrt ( 1 + x2 / ( R2 - x2 ) )
or
2 * pi * sqrt ( R2 - x2 ) * sqrt ( R2 / ( R2 - x2 ) )
or
2 pi R.
Now we do the integral, to find that the surface area is:
2 * pi * R * ( B - A )
In other words, the surface area of the slice depends only on the "thickness" of the slice, and not on where it is taken!

I ran across this fact while actually doing real work. I needed to compute a "random" direction in 3 space. You can do this by picking a random point on a unit sphere.

So how do you pick a random point on a unit sphere? Well, to describe a point on a unit sphere, you only have to specify the two angles theta and phi which specify longitude and latitude respectively. It was pretty clear that theta could just be chosen uniformly between 0 and 2*pi, but how would we weight the choice of phi?

If the points are to be equally likely, then you need to do this in such a way that the points in any patch of the surface have a chance of being picked that is proportional to the area of the patch. Since I was concentrating on phi, my slices were all going to be horizontal bands. I needed to find a relationship between the area of such a band and the change in the angle phi. There's a simple relationship between phi and the height z, namely

z = cos ( phi )
When I kept coming out with the result that the area of band of fixed z width was constant, I just knew this had to be incorrect. I did the integration four times, and then went to lunch, furious and demoralized.

Back to The Slice Puzzle.

Last revised on 06 October 1999.