Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 3

Special Issue: In Honor of Prof. Kyoji Saito’s 75th Birthday

Guest Editors: Stanislaw Janeczko, Si Li, Jie Xiao, Stephen S.T. Yau, and Huaiqing Zuo

On the area formulas of inscribed polygons in classical geometry

Pages: 557 – 572

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n3.a8

Authors

Yohei Komori (Department of Mathematics, School of Education, Waseda University, Shinjuku, Tokyo, Japan)

Runa Umezawa (Department of Mathematics, Faculty of Science and Engineering, Waseda University, Shinjuku, Tokyo, Japan)

Takuro Yasui (Department of Mathematics, School of Education, Waseda University, Shinjuku, Tokyo, Japan)

Abstract

We show that there is no area formula of the general inscribed $n$‑gon for $n \geq 5$ only by using arithmetic operations and $k$‑th roots of its side lengths in classical geometry.

Keywords

Euclidean geometry, hyperbolic geometry, spherical geometry

2010 Mathematics Subject Classification

Primary 51Kxx. Secondary 51M10, 51N20.

Full Text (PDF format)

Received 20 June 2019

Accepted 13 August 2019

Published 11 November 2020