Volume of a Cone or PyramidDate: 03/30/98 at 16:04:10 From: Laura Lee Swan Subject: Volume of cone/pyramid Hi there, I have been happily entertained for the last 40 minutes searching your web site for proof that the volume of cone/pyramid = (1/3)b*h. I found something, but it mentioned Riemann sums, so I stopped. Can you explain the rest? I've been searching for an answer for weeks and can't get my hands on the right text book or person. My superstar 8th grader wants to know why, and as a permanent sub (I am not a permanent math teacher who ought to know how to explain this), I'm stumped. I'm thrilled that he's asking, so I want to be able to give him an answer (no matter how difficult). Thanks. Date: 03/30/98 at 22:55:45 From: Doctor Rob Subject: Re: Volume of cone/pyramid Imagine a step-pyramid made up of small square prisms with dimensions a-by-a-by-b, arranged in square layers. If the layers are k-by-k for k = 1, 2, 3, ..., n, then the total volume of the step-pyramid is: V = a^2*b*[1^2 + 2^2 + 3^2 + ... + n^2], = a^2*b*n*(n + 1)*(2*n + 1)/6 (which can be proved by induction), = a^2*b*n^3*(1/3 + 1/[2*n] + 1/[6*n^2]). Now, the height of the step-pyramid is h = b*n, and the area of the base is: B = a^2*n^2, so the volume is: V = h*B*(1/3 + 1/[2*n] + 1/n^2). Now let n -> infinity, so the steps get smaller and smaller, and the volume of the step-pyramid approaches the volume of the inscribed pyramid. Then the right-hand side approaches h*B/3, as the volume V approaches the volume of the pyramid. This same proof works for a pyramid with any polygonal base. For a cone, take a pyramid with a regular m-gon for a base, and let m -> infinity. The base approaches a circle, and the pyramid approaches a cone, and the formula V = h*B/3 still holds in the limit. There is another proof, which cuts a rectangular solid into six pieces of equal volume, each of which is a triangular pyramid (or irregular tetrahedron). This proves that the volume of each is 1/6th the volume of the starting rectangular solid, and each has base of area 1/2 the area of one of the faces of the solid, and height equal to the dimension of the solid perpendicular to that face. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 01/14/99 at 20:34:06 From: Sandy Deering Subject: Volume of Cone Dear Dr. Math, I read this letter on the Volume of a Cone or Pyramid but I still didn't find my answer. My question is: What does the formula for the volume of a cone (1/3 * pi * r^2 * h) really mean? Why do we use that formula? Date: 01/15/99 at 00:57:39 From: Doctor Pat Subject: Re: Volume of Cone Sandy, The volume of a cone may be thought of as a special case of the volume of a cylinder. The volume of a prism-like object is equal to its base area times its height. That is true for any solid for which each cross section is identical at any height. A box and a cylinder are examples. This is known as Cavalerri's principal. That is where the Pi*r*r comes into the problem. The base area of a cone and cylinder is a circle. If the base on the top is reduced to a similar shape, but smaller size, the volume decreases. It doesn't matter what shape the top and bottom are, this is true for all of them. If we reduce the similar shape at the top to a single point, we make a pyramid or cone. The volume of this new figure is 1/3 the volume of the prism-like form. We need calculus to prove this rigorously. To see how, take a look at: http://mathforum.org/dr.math/problems/swimduck5.29.96.html But to just convince yourself only requires some appropriate cylinders and cones with the same base. Ask your teacher if your school has a set of volume containers in these shapes, and then pour rice from the cone to the cylinder. Three conefuls should fill it right up. - Doctor Pat, The Math Forum http://mathforum.org/dr.math/ |
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