Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-17T23:34:57.519Z Has data issue: false hasContentIssue false

A characterisation of regular n-gons via (in)commensurability

Published online by Cambridge University Press:  15 February 2024

Silvano Rossetto
Centro Morin, Paderno del Grappa, Italy e-mail:
Giovanni Vincenzi
Dipartimento di Matematica, Universita di Salerno, via Giovanni Paolo II, 132, I-84084 Fisciano (SA), Italy e-mail:


In Euclidean geometry, a regular polygon is equiangular (all angles are equal in size) and equilateral (all sides have the same length) polygon. So regular polygons should be thought of as special polygons.

© The Authors, 2024 Published by Cambridge University Press on behalf of The Mathematical Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Barbeau, E. J., Incommensurability proof: a pattern that peters out, Mathematics Magazine, 56(2) (1983) pp. 8290.CrossRefGoogle Scholar
Cooke, R., Life on the mathematical frontier: legendary figures and their adventures. Notices of the American Mathematical Society, 57 (4) (2010) pp. 464474.Google Scholar
Vincenzi, G., Paolillo, B. and Rizzo, P., Commensurable diagonals in regular n-gons, International Journal of Mathematical Education in Science and Technology, 53(10) (2021) pp. 2831–2826/CrossRefGoogle Scholar
Vincenzi, G., A characterization of regular n-gons whose pairs of diagonals are either congruent or incommensurable, Arkiv der Mathematik, 115: (2020) pp. 464477.Google Scholar
Rossetto, S. and Vincenzi, G., Considerazioni sulla commensurabilita nei solidi platonici, L’insegnamento della matematica e delle scienze integrate, 46(B), (2023) pp. 107128.Google Scholar
Vorobev, E. M., Teaching of real numbers by using the Archimedes– Cantor approach and computer algebra systems, International Journal of Mathematical Education in Science and Technology, 46(8) (2015) pp. 11161129.CrossRefGoogle Scholar
Fontaine, A. and Hurley, S., Proof by picture: products and reciprocals of diagonal length ratios in the regular polygon, Forum Geometricorum, 6 (2006) pp. 97101.Google Scholar
Khan, S. A., Trigonometric ratios using geometric methods, Advances in Mathematics: Scientific Journal (9) 10: (2020) pp. 86858702.Google Scholar
Calcut, J. S., Grade school triangles, Amer. Math. Monthly, 117 (2010) pp. 673685.CrossRefGoogle Scholar
Garibaldi, S., Somewhat more than governors need to know about trigonometry, Math. Magazine. 81(3) (2008) pp. 507508.CrossRefGoogle Scholar
Paolillo, B. and Vincenzi, G., On the rational values of trigonometric functions of angles that are rational in degrees, Math. Magazine, 94(2) pp. 132134.CrossRefGoogle Scholar
Paolillo, B. and Vincenzi, G., An elementary proof of Niven’s theorem via the tangent function, International Journal of Mathematical Education in Science and Technology, 52(6) (2021) pp. 959964.CrossRefGoogle Scholar
Conway, J. H. and Guy, R. K., The only rational triangle: in The book of numbers, Springer-Verlag (1996).CrossRefGoogle Scholar
Evans, R. and Isaacs, M. I., Special non-isosceles triangle: solution of the Problem number 2668 Amer. Math. Monthly, 85 (10) (1978) p. 825.CrossRefGoogle Scholar
Parnami, J. C., Agrawal, M. K. and Rajwade, A. R., Triangles and cyclic quadrilaterals, with angles that are rational multiples of π and sides at most quadratic over the rationals, Math. Student, 50 (1982) pp. 7993.Google Scholar