Volume of a ConeDate: 02/08/2002 at 01:09:37 From: Henry Subject: Another method of calculating volume of cone Doctor Math: Regarding the formula for calculating the volume a cone, I have observed that a cone consists of a right angle triangle and a circle base. I have came up with another method of calculating the volume of a cone, which is: (The Area of the Right Angle triangle)*(the base circle circumference) r = radius of base circle h = height of triangle (1/2rh)(2rPi) but solving the above does not give me the formula 1/3Pir^2h. I am wondering if my formula is wrong. Please help correct it. Date: 02/08/2002 at 09:08:08 From: Doctor Peterson Subject: Re: Another method of calculating volume of cone Hi, Henry. Something close to your method works. The problem is that although the volume of a triangle moved in a straight line (a triangular prism) is the product of its area and the distance moved, when the triangle moves in a circle, not all of it moves the same distance, so you can't use the OUTER circumference as if the whole triangle moved that far. But it turns out (and can be proved with some advanced geometry, or with calculus) that the volume of such a "solid of revolution" will be the product of the area of the triangle and the distance moved by the center of gravity (centroid) of the triangle. Since this is 1/3 of the way out from the axis, this multiplies your formula by 1/3 and makes it correct. You can read about this "Theorem of Pappus" at Eric Weisstein's World of Mathematics: http://mathworld.wolfram.com/PappussCentroidTheorem.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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