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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.

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

Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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