Thickness of Ocean Calculation

I wanted to calculate the height of water above a perfectly smooth earth sphere if all the water on earth surrounded the sphere. I found the volume of all water on line to be 322.5 cu miles. Using a 400mile radius for my earth sphere, I calculated the volume of this sphere (268.082E9 cu miles) and then added the 322.5E6 cu miles of water and solved for the difference in the two radii (with and without surrounding water). Answer 1.6 miles. All great, but then a friend did the calculation by dividing the surface area of my earth sphere (201.062E6 sq miles into the volume of water (322.5E6 cu miles) and got the same answer--1.6 mile high. He claims you can use the rectangular volume calculation and get the same answer as with the spherical volume calculation because somehow the earth's gravitational system equates the two. I don't get it and think he is blowing smoke, but I can't figure out why I get the same answer using both methods (spherical and rectangular). ANY HELP?
- CHARLES MCCARTHY (age 75)
BOW, WASHINGTON, USA

Your method is correct- solve for the radii of the bigger and smaller spheres and take the difference. Your friend's method works here too but only because the extra volume is such a small fraction of the total. As a result, the outer and inner areas of the extra layer are almost exactly equal. So you can get the layer's volume by multiplying its thickness by the inner area.Your friend's approximation is so good here that it's the easier way to do the calculation, because you don't have to precisely calculate two big numbers (the radii) and then accurately take the difference.  If the layer were thicker, those inner and outer areas wouldn't be equal and his method wouldn't work. Gravity has nothing directly to do with calculation. 

Mike W.

(published on 09/19/2015)