Volume of a Cone

Date: 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/