Formula for the Surface of a Cylinder

Date: Tue, 2 Jul 1996 13:47:47 -0400 (EDT) 
From: Anonymous
Subject: General cylinder surface formula 

Hi,

My son studied (amongst other things) cylinders in his math class 
last semester. The class covered the formula for a right cylinder, 
but my son is curious to know the general formula for cylinders in 
any orientation. In other words, given two three-dimensional end 
points (x0, y0, z0) and (x1, y1, z1), and the radius of the cylinder, 
what would be the resulting formula to describe the surface of the 
cylinder?

I've looked around a lot on the Internet, but I haven't been able to 
find the general formula. We figure it must be similar to the sphere 
formula x*x + y*y + z*z = r*r. Also, we think that the equation 
will be the equivalent of an infinite number of circles, all 
perpendicular to and centered around the line from the first point 
to the second. 

We'd certainly appreciate any help you can offer. Thanks! 

Graham MacRobie (for Andrew MacRobie)

Date: Tue, 2 Jul 1996 18:06:43 -0400 (EDT) 
From: Dr. Anthony
Subject: Re: General cylinder surface formula 

If you slice through a circular cylinder in a direction which is not 
perpendicular to the axis, then the shape of the figure produced 
will be an ellipse.

If therefore we have a cylinder with its two end planes not at right 
angles to the axis, then the curved surface area would be the 
perimeter of the ellipse multiplied by the perpendicular height. 
Since calculating the perimeter of an ellipse involves a particularly 
difficult integral, it is certain that there will be no convenient 
formula for the surface area. The volume, on the other hand would 
be relatively easy to calculate, using the formula pi*a*b for the 
cross-section area of the ellipse, where 2a and 2b are the major and 
minor axes of the ellipse. 

-Doctor Anthony, The Math Forum