Lateral Areas of Right and Oblique Circular Cylinders

Date: 06/03/2002 at 17:32:51
From: P. McGrath
Subject: Derivation for the surface area of an oblique cylinder

I was wondering if you could explain to me why the lateral surface 
area of an oblique cylinder is calculated using the slant height
(length of a lateral edge) multiplied by the circumference of the 
circular base.  

Intuitively it doesn't make sense to me that an oblique cylinder's
lateral area would be calculated using the same measurements as a
right cylinder's lateral area.  I can visualize the lateral area of a
right cylinder opening up to be a rectangle, but am having a hard time
seeing this with the lateral area of an oblique 
cylinder.  

Thanks,

P. McGrath

Date: 06/04/2002 at 13:40:37
From: Doctor Peterson
Subject: Re: Derivation for the surface area of an oblique cylinder

Thanks for writing!

Actually, the lateral surface area is NOT the product of the slant 
height and the circumference of the circular base. Looking at our FAQ

    http://mathforum.org/dr.math/faq/formulas/faq.cylinder.html

I at first thought that was what it said, but then I looked closer:

     Height: h
     Area of base: B
     Length of lateral edge: l
     Area of right section: A
     Perimeter of right section: P

     Lateral surface area: S
     Total surface area: T
     Volume: V

          S = lP
          T = lP + 2B
          V = hB = lA

Note that "P" is not defined as "perimeter of base", but as 
"perimeter of right section", just as "A" is "area of right section" 
in contrast to B, the "area of base". 

Neither A nor P is shown on the diagram or explained carefully; a
right section is a cross section _perpendicular to the axis_, not
_parallel to the base_. 

Picture making an oblique cylinder as a bundle of sticks with length
l, then standing the cylinder so that the axis is vertical, and
pressing down so that all the sticks (vertical generators of the
cylinder) touch the table. Then you will have an elliptical cylinder
with height l and base area A; its volume, as indicated, will be lA
(unchanged). 

Its lateral surface area will be lP, where P is the perimeter of the 
new elliptical base, and this too will be unchanged from the original 
oblique cylinder.

So (with this very intuitive "proof") the formulas given here are 
correct, as long as you understand that P is not the perimeter of the 
base of the oblique cylinder. I think this should be more clearly 
stated on the page, so it could not be misread. If you got this 
formula from elsewhere, and P was defined as the circumference of the 
base, then you are right in questioning it.

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