416. Volumes, Surfaces Areas, and Centroids of Prismoidal and Revolved Surfaces

Paper

Abstract

At times, considerable improvement in calculation results can be obtained by a more thorough understanding of, and the proper use of existing calculation methods, new ‘short-cut’ methods and approximations. Slight rearrangement of existing formulas make them much easier to use, and applicable to more situations
At times, considerable improvement in calculation results can be obtained by a more thorough understanding of, and the proper use of existing calculation methods, new ‘short-cut’ methods and approximations. Slight rearrangement of existing formulas make them much easier to use, and applicable to more situations that occur in weight work. When to use, and when not; to use, an ‘End Averaging Method’ in lieu of a ‘Prismoidal Method’, is often a problem, depending on the accuracy desired. This paper goes into this problem, and gives examples.
This paper not only presents in one place, useful formulas and graphs, but by development, illustration, and discussion goes into ‘Prismoidal Methods’, and Revolved Shells, and Solids detail, so that a comprehensive, and instructive guide is available for weight personnel. Both mathematical, and pictorial proofs are presented for easy understanding. Examples are given.
The paper is presented in order to help weight and design people improve the results of some of the most difficult volume, area, and centroidal calculations, in less time. It is also written in order to bring about a better understanding, and use, of some formulas not yet found any of the S.A.W.E. weight handbooks. These formulas, calculation methods, and graphs, with examples of their use, are given.
Development of the general ‘Prismoidal Formula’, ‘Prismoid Formula’, and ‘Lateral Surface Area Formula’ have been placed in Appendix I so that the body of the paper would be easier to read. Related useful formulas are contained in Appendix II for the same reason. Also a special page, giving a direct comparison of the ‘Prismoidal Method’ to the ‘End Averaging Method’, is given in Appendix II, where it can be easily removed and used as a’Designer-Reminder’ page. Reproductions of this one page maybe very useful to the weight engineer, if passed out to the design group before they are required to make a weight calculation involving a Prismatoid.