1. bookVolume 29 (2021): Issue 2 (June 2021)
Journal Details
License
Format
Journal
eISSN
1844-0835
First Published
17 May 2013
Publication timeframe
1 time per year
Languages
English
access type Open Access

A Generalization of Archimedes’ Theorem on the Area of a Parabolic Segment

Published Online: 08 Jul 2021
Page range: 199 - 209
Received: 24 Sep 2020
Accepted: 15 Nov 2020
Journal Details
License
Format
Journal
eISSN
1844-0835
First Published
17 May 2013
Publication timeframe
1 time per year
Languages
English
Abstract

Archimedes’ well known theorem on the area of a parabolic segment says that this area is 4/3 of the area of a certain inscribed triangle. In this paper we generalize this theorem to the n-dimensional euclidean space, n ≥ 3. It appears that the ratio of the volume of an n-dimensional solid bounded by an (n − 1)-dimensional hyper-paraboloid and an (n − 1)-dimensional hyperplane and the volume of a certain inscribed cone (we analogously repeat Archimedes’ procedure) depends only on the dimension of the euclidean space and it equals to 2n/(n +1).

Keywords

MSC 2010

[1] L. E. Blumenson, A derivation of n-Dimensional Spherical Coordinates, The American Mathematical Monthly, Vol. 67, No. 1 (1960), pp. 63-66. Search in Google Scholar

[2] T. Dance, G. Swain, Archimedes’ Quadrature of Parabola Revisited, Mathematics Magazine, Vol. 71, No. 2 (1998), pp. 123-130. Search in Google Scholar

[3] T. L. Heath, The Works of Archimedes, C. J. Clay and Sons, London, 1897. Search in Google Scholar

[4] W. Rudin, Principles of the Mathematical Analysis, third edition, McGraw-Hill, Inc., New York, 1976. Search in Google Scholar

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