Surface Area and Volume of a SphereDate: 05/16/2000 at 22:31:39 From: Randy S. Subject: Surface Area and Volume of a Sphere Hello, I am doing research for a project on why the coefficient in the formula for the surface area of a sphere is 4, and why 4/3 is the coefficient in the formula for the volume of a sphere. I have looked through your archives and have found that you can prove both by using an infinite number of pyramids inside a sphere, but is that the only geometric way? All the other methods used in the archives are too complicated for me (some use Calculus.) I also have some trouble understanding Archimedes' hatbox because I don't understand why they multiply cosine and latitude lines. I also don't understand how one person proved it by using integrals because I don't even know what an integral is. So could you explain it to me geometrically, or else try to find some other geometric ways to explain why the coefficients are 4 and 4/3? Date: 05/16/2000 at 23:15:55 From: Doctor Peterson Subject: Re: Surface Area and Volume of a Sphere Hi, Randy. Don't worry about the integrals; that's calculus, and you'll need to learn a bit before you can follow the whole argument. The geometrical methods really use some of the ideas of calculus, but not the methods of calculus, so they can be followed more easily, though the work is harder. It's not clear to me whether you saw this answer, which I think is the same as the hat-box, but doesn't mention cosines and latitudes, so it may be easier to follow: Volume of a Sphere http://mathforum.org/dr.math/problems/banijamal05.28.99.html There I start by finding the surface area using that method, and then use the pyramids to get the volume from that. There's another method that gets the volume directly using Cavalieri's theorem, which says that if every cross-section of two solids has the same area, then they have the same volume. We show that the sphere has the same volume as the cylinder circumscribed about the sphere, with a cone removed from each end. See Volume of a Hemisphere Using Cavalieri's Theorem http://mathforum.org/dr.math/problems/pappas.9.09.99.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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