Volume of a Prismatoid
Date: 10/21/2008 at 10:05:49 From: Ravi Subject: Volume of circular end pyramid How can I calculate the volume of a pyramid for which the bottom end is rectangular and the top end is circular? I'm totally confused.
Date: 10/21/2008 at 10:49:23 From: Doctor Peterson Subject: Re: Volume of circular end pyramid Hi, Ravi. This is not really a pyramid, but I picture it as a sheet-metal transition from a round pipe to a rectangular duct, which is formed by bending a series of triangles with one base in one end of the figure, and the vertex in the other. Is that right? The volume can be calculated using the formula for a prismatoid, which is the same kind of shape except that the top is a polygon rather than a circle: Prismatoid http://mathworld.wolfram.com/Prismatoid.html The formula is V = h/6 (A1 + 4M + A2) where A1 and A2 are the top and bottom areas, and M is the midsection area. In our case, if r = radius of circle L = length of rectangle W = width of rectangle it turns out that A1 = pi r^2 A2 = LW M = (L/2 + r)(W/2 + r) - r^2 + pi r^2/4 and so V = h/3 (pi r^2 + LW + Lr + Wr) That is such a nice formula, I imagine I am not the first to write it! See if this works for you. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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