Volume of n-dimensional Spheres

Tairan Zhong; Jayne Jiang; Tina Zhang

doi:10.54254/2753-8818/2025.22642

Research Article

Open access

Published on 6 May 2025

Download pdf

Zhong,T.;Jiang,J.;Zhang,T. (2025). Volume of n-dimensional Spheres. Theoretical and Natural Science,108,70-77.

Export citation

Volume of n-dimensional Spheres

Tairan Zhong *^,1, Jayne Jiang ², Tina Zhang ³

¹ Guangdong Country Garden School
² New channel
³ New channel

* Author to whom correspondence should be addressed.

https://doi.org/10.54254/2753-8818/2025.22642

Abstract

In this paper, we present two methods to derive the formula for the volume of n-dimensional spheres in Euclidean space and analyze its asymptotic behavior as n approaches infinity using Stirling’s formula. We then establish a connection between the infinite sum of all even-dimensional spheres of radius r and the exponential function through a differential equation. Our findings highlight the decay characteristics of high-dimensional volumes and reveal a novel link between geometry and analysis. This work not only refines classical volume estimation techniques but also offers valuable insights into applications in higher-dimensional mathematics, contributing to fields such as statistical mechanics and theoretical physics.

Keywords

n-dimensional spheres, Euclidean space, Stirling’s formula, asymptotic behavior, high-dimensional volumes

View pdf

References

[1]. Physics 2400. 2017. “The surface area and the volume of n-dimensional sphere”. Spring Semester 2017. http://www.phys.uconn.edu/˜rozman/Courses/P2400_17S/

[2]. FengZhuYunMo. 2021. “The formula for the volume of an n-dimensional hypersphere”. Bilibili. https://b23.tv/BFMeBB2

[3]. Sanderson. (2023). “Why pi appear in the normal distribution, YouTube. Available at: https://www.youtube.com/watch?v=cy8r7WSuT1I&t=632s (Accessed: 15 September 2024).

[4]. Functor7 (2016) R/math on reddit: Why, intuitively, does the sum of the volumes of even dimension balls equal e^pi?, Why, intuitively, does the sum of the volumes of even dimension balls equal e^pi? Available at: https://www.reddit.com/r/math/comments/4qryk6/why_intuitively_does_the_sum_of_the_volumes_of/ (Accessed: 15 September 2024)

[5]. Sujianlin (23 December 2014): Find the volume of an n-dimensional sphere. https://spaces.ac.cn/archives/3154

[6]. Avrim Blum､John Hopcroft (9 February 2016): The Volume of High-dimensional Unit Ball https://freemind.pluskid.org/misc/the-volume-of-high-dimensional-unit-ball/

[7]. yingqizhi (4 November 2016) A review of derivation methods for the volume formula of an n-dimensional sphere.https://m.fx361.com/news/2016/1104/19014426.html

[8]. addis Volume of multi-dimensional sphere. https://wuli.wiki/NSphV

[9]. University of Connecticut - n-Dimensional Sphere Surface Area and Volumehttps://www.reddit.com/r/math/comments/4qryk6/why_intuitively_does_the_sum_of_the_volumes_of/?rdt=57019

[10]. Blue1Brown: Why π appears in the normal distribution https://www.youtube.com/watch?v=cy8r7WSuT1I&t=632s

Cite this article

Zhong,T.;Jiang,J.;Zhang,T. (2025). Volume of n-dimensional Spheres. Theoretical and Natural Science,108,70-77.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

Disclaimer/Publisher's Note

The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of EWA Publishing and/or the editor(s). EWA Publishing and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

About volume

Volume title: Proceedings of the 4th International Conference on Computing Innovation and Applied Physics

Conference website: https://2025.confciap.org/

ISBN：978-1-80590-089-4(Print) / 978-1-80590-090-0(Online)

Conference date: 17 January 2025

Editor：Ömer Burak İSTANBULLU, Marwan Omar, Anil Fernando

Series: Theoretical and Natural Science

Volume number: Vol.108

ISSN：2753-8818(Print) / 2753-8826(Online)

© 2024 by the author(s). Licensee EWA Publishing, Oxford, UK. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license. Authors who publish this series agree to the following terms:
1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this series.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this series.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See Open access policy for details).