Derivation of Sphere Volume and Surface Area Formulas
Date: 03/14/2005 at 16:10:28 From: Alex Subject: Why the formula for the volume of a Sphere has '4/3' in it Currently in math class we are discussing surface areas and volumes of solids. I would like to know why the volume formula for the sphere is (4/3)*pi*r^3 and why the surface area formula is 4*pi*r^2. I understand why the volume formula contains pi, and radius, and is cubed, but I do not understand why 4/3 must be in the formula. The same goes for the surface area formula. I can understand the pi*r^2, but I do not understand why it is 4 and not another number. I'm sure that both the (4/3) and the 4 were derived some way, but I cannot understand why it is so. I often question formulas like this.
Date: 03/15/2005 at 08:23:21 From: Doctor Rick Subject: Re: Why the formula for the volume of a Sphere has '4/3' in it Hi, Alex. I appreciate your desire to understand the reasons for things and not just take a teacher's word for it. That is an important trait in a mathematician. Unfortunately, it will be a while before you have enough math under your belt to understand the proof in full detail. It involves calculus. For now, we can help you see that it makes sense, though, even if we can't show that it *must* be true. Let's start with the surface area. Take a sphere of radius R, and imagine constructing a cylindrical box with radius R and height 2R. You can see that the sphere will fit snugly inside this box. Archimedes, the Greek mathematician, proved a surprising fact: the surface area of the sphere is exactly the same as the lateral surface area of the cylinder (that is, the surface area not including the two circular ends). You can calculate the lateral surface area of the cylinder and you will see that it is 4*pi*R^2. The following item in the Dr. Math Archives describes what Archimedes did to prove this result: Volume of a Sphere http://mathforum.org/library/drmath/view/55135.html Also on that page you will see an explanation of the 4/3 in the volume of the sphere. In brief, you can imagine drawing a tiny triangle on the surface of the sphere and connecting its corners to the center of the sphere. You have made a very narrow pyramid. The volume of a pyramid is 1/3 times the area of the base times the height. Thus the volume of this pyramid is 1/3 times the radius of the sphere, times the area of that little triangle. Now, imagine that you cover the sphere with tiny triangles, and thus cut the sphere into millions of narrow pyramids. The total volume of the pyramids is 1/3 times the radius of the sphere times the sum of the areas of the tiny triangles. In other words, the volume of the sphere is 1/3 times R times the surface area of the sphere! V = (1/3)R * 4*pi*R^2 = (4/3)pi*R^3 How's that? - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 03/15/2005 at 19:39:00 From: Alex Subject: Why the formula for the volume of a Sphere has '4/3' in it Thank you for your answer. Unfortunately, I believe that there is an error. When I looked up the formula for the lateral area of a cylinder, I found that it is 2*pi*r*h or pi*d*h. The formula that you used to explain why it is (4/3)*pi*r^3 contains the surface area of a sphere, not a cylinder, in it. The formula makes sense, and so does the explanation why, but it is confusing that the surface area of the sphere is interchanged with that of the cylinder. As far as I understand, this is impossible. I thank you in advance for your time.
Date: 03/16/2005 at 08:15:18 From: Doctor Rick Subject: Re: Why the formula for the volume of a Sphere has '4/3' in it Hi, Alex. Didn't I say this fact was "surprising"? I'm glad you agree! To me, it's pretty amazing. In fact, Archimedes thought it was so wonderful that he had a carving of a sphere and a cylinder put on his tombstone. The surface area of a sphere is 4*pi*r^2. The lateral area of a cylinder is the height times the circumference of the base, or 2*pi*r*h. For a cylinder of height 2r, that makes S = 2*pi*r*(2r) = 4*pi*r^2 Whether you think it's impossible or not, it's true! The surface area of a sphere is the same as the lateral surface area of the cylinder into which the sphere fits. I pointed you to a page where we describe Archimedes' method of proving this: Volume of a Sphere http://mathforum.org/library/drmath/view/55135.html That page, like my explanation, begins with the surface area, then uses that to show that the formula for volume of a sphere makes sense. Doctor Peterson (who happens to be my twin brother) talks about slicing through both the sphere and the cylinder to make lots of very thin slices. Then he demonstrates that the surface area of each slice of the sphere is equal to the surface area of the corresponding slice of the cylinder. Though the radius of the sphere slice is less than the radius of the cylinder slice, the surface of the sphere slice is sloped, which makes it longer (measuring along the slope) than the surface of the cylinder slice. These two effects happen to cancel out: the factor by which the radius is reduced is the reciprocal of the factor by which the length is increased, so the surface areas of the two slices come out the same. And if the areas of the slices are the same, then the areas of the whole sphere and the whole cylinder (without the ends) are the same. You'll have to go through Doctor Peterson's explanation to see that this is true. I can't make it any simpler than that. I'll be glad to answer more questions, though, until you're satisfied. Just remember that *full* satisfaction won't come until you get to calculus. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 03/16/2005 at 14:58:54 From: Alex Subject: Thank you (Why the formula for the volume of a Sphere has '4/3' in it) Thank your very much for your help. I now understand where the surface area of a cylinder and sphere can be the same. I was confused because I thought that you were saying that the surface area of a cylinder and sphere is always exactly the same. Now I understand. Thank you very much! Sincerely, Alex
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