Lateral Areas of Right and Oblique Circular CylindersDate: 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/ |
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