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Area, Volume of a Cone


Date: 6/4/96 at 19:52:50
From: T.J. Jenkins
Subject: Area, Volume of a Cone

What are the formulae to find the volume and area of a cone?


Date: 6/5/96 at 18:47:35
From: Doctor Charles
Subject: Re: Area, Volume of a Cone

The volume of a cone with base radius r and height h is:

         V = 1/3 * pi * r^2 *h

The only proof that I can remember uses calculus. In fact you can show 
that if you have any tube shape (triangular prism etc.) and form the 
'pointed' version (triangle based pyramid etc.) then the volume is a 
third of what you started with.

You can make the curved surface of a cone by rolling up a sector of a 
circle.

This gives that the curved surface has surface area:

    Area = pi * (h^2 + r^2) * [2 * pi * r] / [2 * pi * sqrt(h^2 +r^2)]
         = pi * sqrt (h^2 + r^2) *r

The area of the bottom circle is of course pi * r^2 so the total 
surface area of the cone is:

     Area = pi * r * [sqrt (h^2 + r^2) + r]

-Doctor Charles,  The Math Forum
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Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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