Surface Area of Cones and Pyramids

Date: 09/27/2003 at 21:41:33
From: Nadiya
Subject: Surface area of cones and pyramids

Can the method for finding the surface area of a pyramid be used as
well to find the surface area of a cone? 

Date: 09/28/2003 at 10:40:46
From: Doctor Jerry
Subject: Re: Surface area of cones and pyramids

Hi Nadiya,

Yes, if you're willing to be a bit loose with procedures.

The area of pyramid is 2*a*s,  where a is the side of the base and s
the slant height.  I suppose one can find this by looking at one
triangle, whose area is 

  (1/2) * height * base = (1/2)*s*(a)

There are four of these, for a total area of 2*a*s.

On the cone, divide the circle into a large number of equal pieces,
say n of them.  We know that each has a base of length 2*pi*r/n.  So,
this is a.  The slant height is s.  Each base can be connected to the
vertex, making a "triangle."  The resulting area is

  (1/2) * s * 2*pi*r/n   

There are n of these for a total area of pi*r*s.

I believe that Kepler used these kinds of (risky) arguments.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/