If you have a truncated cone such that its upper base has a radius of a and the radius of its [larger] lower base is b, and a height h (between bases), how could you figure out its surface area using geometric reasoning?

Side + Top + Bottom

Pi*(a+b)* sqrt[h^2 + (b-a)^2] + Pi*a^2 + Pi*b^2

The geometric explanation for the top and bottom is obvious.
For the geometric equation of the side area, consider a cylindar instead of a truncated cone (is this called a frustum?).  Anyway, consider a cylindar of radius (a+b)/2.  The side area of that would be h*2*Pi*[(a+b)/2]  =  h*Pi*(a+b)

But for the truncated cone, we need the slant height, S, which is the hypoteneuse of a right triangle with sides (h) and (b-a).

So a cylindar whose radius is the average of the top and bottom of the truncated cone; and whose height is the same as the slant height of the truncated cone should have the same side area.

And that's my story.

Posted by Larry on 2004-03-15 01:16:55