<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Archimedes’ well known theorem on the area of a parabolic segment says that this area is 4<italic>/</italic>3 of the area of a certain inscribed triangle. In this paper we generalize this theorem to the <italic>n</italic>-dimensional euclidean space, <italic>n</italic> ≥ 3. It appears that the ratio of the volume of an <italic>n</italic>-dimensional solid bounded by an (<italic>n</italic> − 1)-dimensional hyper-paraboloid and an (<italic>n</italic> − 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 2<italic>n/</italic>(<italic>n</italic> +1).</p>
</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).

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

The John Paul II Catholic University of Lublin ul

Department of Applied Mathematics and Computer Science

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"A Generalization of Archimedes\u2019 Theorem on the Area of a Parabolic Segment"}

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...

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

Publicado en línea: 08 jul 2021

Páginas: 199 - 209

Recibido: 24 sept 2020

Aceptado: 15 nov 2020

DOI: https://doi.org/10.2478/auom-2021-0026

Palabras clave
Archimedes’ theorem, parabolic segment, multiple integral, spherical coordinates

© 2021 Armen Grigoryan et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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

Publicado en línea: 08 jul 2021

Páginas: 199 - 209

Recibido: 24 sept 2020

Aceptado: 15 nov 2020

DOI: https://doi.org/10.2478/auom-2021-0026

Palabras claveArchimedes’ theorem, parabolic segment, multiple integral, spherical coordinates

© 2021 Armen Grigoryan et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Palabras clave
Archimedes’ theorem, parabolic segment, multiple integral, spherical coordinates