Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon

Nenad Stojanovic

doi:doi:10.11648/j.ajam.20200804.12

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Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon

Nenad Stojanovic

Published in American Journal of Applied Mathematics (Volume 8, Issue 4)

Received: 13 May 2020 Accepted: 8 June 2020 Published: 17 June 2020

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Abstract

A simple polygon that either has equal all sides or all interior angles is called a semi-regular polygon. In terms of this definition, we can distinguish between two types of semi-regular polygons: equilateral polygons (that have equal all sides and different interior angles) and equiangular polygons (that have equal interior angles and different sides). To analyze the metric properties of semi-regular polygons, knowing only one basic element, e.g. the length of a side, as in regular polygons, is not enough. Therefore, in addition to the side of a semi-regular polygon, we use another characteristic element of it to analyze the metric features, and that is the angle δ=∠(a,b) between the side of a semi-regular polygon P_{_N} and the side b of its inscribed regular polygon P_{_N}. Some metric properties of a semi-regular equilateral 2n-sides polygon are analyzed in this paper with respect to these two characteristic elements. Some of the problems discussed in the paper are: convexity, calculation of surface area, dependence on the length of sides a and δ, calculation of the radius of the inscribed circle depending on the sides a and angles δ, and calculation of the surface area in which the radius of the inscribed circle is known, as well as the relationship between them. It has been shown that the formula for calculating the surface area of regular polygons results from the formula for the surface area of 2n-side semi-regular, equilateral polygons. Further, by using these results, it has been shown that the cross-sections of regular polygons inscribed to semi-regular equilateral polygons, the vertices of equiangular semi-regular polygons, as well as the sides of the regular polygons inscribed to it, intersect in the same manner at the vertices of the equilateral semi-regular polygon. It has further been shown that the sides of the equiangular semi-regular polygon refer to each other as the sines of the angles created by the sides of the inscribed polygons and the side of the semi-regular polygon.

Published in	American Journal of Applied Mathematics (Volume 8, Issue 4)
DOI	10.11648/j.ajam.20200804.12
Page(s)	176-181
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Semi-Regular Polygons, Surface Ratio, Equilateral and Equiangular Semi-Regular Polygons

References

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[3]	M. Audin, Geometry, Springer, Heidelberg, 2002.
[4]	M. Polonio, D. Crnokvić, T. B. Kirigan, Z. Franušić, R. Sušanj, Euklidski prostori, PMF, Zagreb, 2008, pp. 51-57.
[5]	Kirilov, O pravilnih mnogougolnikah, funkciji Eulera i ćisla Ferma, Kvant, N₀ 6, 1994.
[6]	M. Panov, A. Spivak, Vpisanie poligoni, Kvant, N₀ 1, 1999.
[7]	M. Radojčić, Elementarna Geometrija, Naučna knjiga, Beograd, 1961.
[8]	N. Stojanović, Some metric properties of general semi-regular polygons, Global Journal of Advanced Research on Classical and Modern Geometries, Vol. 1, Issue 2, 2012, pp. 39-56.
[9]	N. Stojanović, Inscribed circle of general semi-regular polygon and some of its features, International Journal of Geometry, Vol. 2., 2013, N₀. 1, pp. 5-22.
[10]	N. Stojanović, V. Govedarica, Jedan pristup analizi konveksnosti i računanju površine jednakostranih polupravillnih poligona, II MKRS, Zbornik radova, Trebinje, 2013, pp. 87-105.
[11]	N. Stojanović, Neka metrička svojstva polupravilnih poligona, Filozofski fakultet Pale, 2015, disertacija.
[12]	N. Stojanović, V. Govedarica, Diofantove jednačine i parketiranje ravni polupravilnim poligonima jedne vrste, Fourth mathematical conference of the Republic of Srpska, Proceedings, Volume I, Trebinje, 2015, pp. 183-194.
[13]	N. Stojanović, V. Govedarica, Diofantove jednačine i parketiranje ravni polupravilnim poligonima dvije vrste, Sixth mathematical conference of the Republic of Srpska, Proceedings, Pale, 2017, pp. 266-280.
[14]	V. V. Vavilov, V. A. Ustinov, Okružnost na rešetkah, Kvant, N₀ 6, 2007.
[15]	V. V. Vavilov, V. A. Ustinov, Mnogougolniki na rešetkah, Izdavateljstvo, MCIMO, Moskva, (2006).

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Nenad Stojanovic. (2020). Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon. American Journal of Applied Mathematics, 8(4), 176-181. https://doi.org/10.11648/j.ajam.20200804.12

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Nenad Stojanovic. Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon. Am. J. Appl. Math. 2020, 8(4), 176-181. doi: 10.11648/j.ajam.20200804.12

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AMA Style

Nenad Stojanovic. Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon. Am J Appl Math. 2020;8(4):176-181. doi: 10.11648/j.ajam.20200804.12

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@article{10.11648/j.ajam.20200804.12,
  author = {Nenad Stojanovic},
  title = {Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {4},
  pages = {176-181},
  doi = {10.11648/j.ajam.20200804.12},
  url = {https://doi.org/10.11648/j.ajam.20200804.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200804.12},
  abstract = {A simple polygon that either has equal all sides or all interior angles is called a semi-regular polygon. In terms of this definition, we can distinguish between two types of semi-regular polygons: equilateral polygons (that have equal all sides and different interior angles) and equiangular polygons (that have equal interior angles and different sides). To analyze the metric properties of semi-regular polygons, knowing only one basic element, e.g. the length of a side, as in regular polygons, is not enough. Therefore, in addition to the side of a semi-regular polygon, we use another characteristic element of it to analyze the metric features, and that is the angle δ=∠(a,b) between the side of a semi-regular polygon PN and the side b of its inscribed regular polygon PN. Some metric properties of a semi-regular equilateral 2n-sides polygon are analyzed in this paper with respect to these two characteristic elements. Some of the problems discussed in the paper are: convexity, calculation of surface area, dependence on the length of sides a and δ, calculation of the radius of the inscribed circle depending on the sides a and angles δ, and calculation of the surface area in which the radius of the inscribed circle is known, as well as the relationship between them. It has been shown that the formula for calculating the surface area of regular polygons results from the formula for the surface area of 2n-side semi-regular, equilateral polygons. Further, by using these results, it has been shown that the cross-sections of regular polygons inscribed to semi-regular equilateral polygons, the vertices of equiangular semi-regular polygons, as well as the sides of the regular polygons inscribed to it, intersect in the same manner at the vertices of the equilateral semi-regular polygon. It has further been shown that the sides of the equiangular semi-regular polygon refer to each other as the sines of the angles created by the sides of the inscribed polygons and the side of the semi-regular polygon.},
 year = {2020}
}

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TY  - JOUR
T1  - Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon
AU  - Nenad Stojanovic
Y1  - 2020/06/17
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200804.12
DO  - 10.11648/j.ajam.20200804.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 176
EP  - 181
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200804.12
AB  - A simple polygon that either has equal all sides or all interior angles is called a semi-regular polygon. In terms of this definition, we can distinguish between two types of semi-regular polygons: equilateral polygons (that have equal all sides and different interior angles) and equiangular polygons (that have equal interior angles and different sides). To analyze the metric properties of semi-regular polygons, knowing only one basic element, e.g. the length of a side, as in regular polygons, is not enough. Therefore, in addition to the side of a semi-regular polygon, we use another characteristic element of it to analyze the metric features, and that is the angle δ=∠(a,b) between the side of a semi-regular polygon PN and the side b of its inscribed regular polygon PN. Some metric properties of a semi-regular equilateral 2n-sides polygon are analyzed in this paper with respect to these two characteristic elements. Some of the problems discussed in the paper are: convexity, calculation of surface area, dependence on the length of sides a and δ, calculation of the radius of the inscribed circle depending on the sides a and angles δ, and calculation of the surface area in which the radius of the inscribed circle is known, as well as the relationship between them. It has been shown that the formula for calculating the surface area of regular polygons results from the formula for the surface area of 2n-side semi-regular, equilateral polygons. Further, by using these results, it has been shown that the cross-sections of regular polygons inscribed to semi-regular equilateral polygons, the vertices of equiangular semi-regular polygons, as well as the sides of the regular polygons inscribed to it, intersect in the same manner at the vertices of the equilateral semi-regular polygon. It has further been shown that the sides of the equiangular semi-regular polygon refer to each other as the sines of the angles created by the sides of the inscribed polygons and the side of the semi-regular polygon.
VL  - 8
IS  - 4
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Nenad Stojanovic

Department of Mathematics, Faculty of Agriculture, University of Banja Luka, Banja Luka, Bosnia and Herzegovina

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Nenad Stojanovic. (2020). Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon. American Journal of Applied Mathematics, 8(4), 176-181. https://doi.org/10.11648/j.ajam.20200804.12

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ACS Style

Nenad Stojanovic. Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon. Am. J. Appl. Math. 2020, 8(4), 176-181. doi: 10.11648/j.ajam.20200804.12

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AMA Style

Nenad Stojanovic. Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon. Am J Appl Math. 2020;8(4):176-181. doi: 10.11648/j.ajam.20200804.12

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@article{10.11648/j.ajam.20200804.12,
  author = {Nenad Stojanovic},
  title = {Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {4},
  pages = {176-181},
  doi = {10.11648/j.ajam.20200804.12},
  url = {https://doi.org/10.11648/j.ajam.20200804.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200804.12},
  abstract = {A simple polygon that either has equal all sides or all interior angles is called a semi-regular polygon. In terms of this definition, we can distinguish between two types of semi-regular polygons: equilateral polygons (that have equal all sides and different interior angles) and equiangular polygons (that have equal interior angles and different sides). To analyze the metric properties of semi-regular polygons, knowing only one basic element, e.g. the length of a side, as in regular polygons, is not enough. Therefore, in addition to the side of a semi-regular polygon, we use another characteristic element of it to analyze the metric features, and that is the angle δ=∠(a,b) between the side of a semi-regular polygon PN and the side b of its inscribed regular polygon PN. Some metric properties of a semi-regular equilateral 2n-sides polygon are analyzed in this paper with respect to these two characteristic elements. Some of the problems discussed in the paper are: convexity, calculation of surface area, dependence on the length of sides a and δ, calculation of the radius of the inscribed circle depending on the sides a and angles δ, and calculation of the surface area in which the radius of the inscribed circle is known, as well as the relationship between them. It has been shown that the formula for calculating the surface area of regular polygons results from the formula for the surface area of 2n-side semi-regular, equilateral polygons. Further, by using these results, it has been shown that the cross-sections of regular polygons inscribed to semi-regular equilateral polygons, the vertices of equiangular semi-regular polygons, as well as the sides of the regular polygons inscribed to it, intersect in the same manner at the vertices of the equilateral semi-regular polygon. It has further been shown that the sides of the equiangular semi-regular polygon refer to each other as the sines of the angles created by the sides of the inscribed polygons and the side of the semi-regular polygon.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon
AU  - Nenad Stojanovic
Y1  - 2020/06/17
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200804.12
DO  - 10.11648/j.ajam.20200804.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 176
EP  - 181
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200804.12
AB  - A simple polygon that either has equal all sides or all interior angles is called a semi-regular polygon. In terms of this definition, we can distinguish between two types of semi-regular polygons: equilateral polygons (that have equal all sides and different interior angles) and equiangular polygons (that have equal interior angles and different sides). To analyze the metric properties of semi-regular polygons, knowing only one basic element, e.g. the length of a side, as in regular polygons, is not enough. Therefore, in addition to the side of a semi-regular polygon, we use another characteristic element of it to analyze the metric features, and that is the angle δ=∠(a,b) between the side of a semi-regular polygon PN and the side b of its inscribed regular polygon PN. Some metric properties of a semi-regular equilateral 2n-sides polygon are analyzed in this paper with respect to these two characteristic elements. Some of the problems discussed in the paper are: convexity, calculation of surface area, dependence on the length of sides a and δ, calculation of the radius of the inscribed circle depending on the sides a and angles δ, and calculation of the surface area in which the radius of the inscribed circle is known, as well as the relationship between them. It has been shown that the formula for calculating the surface area of regular polygons results from the formula for the surface area of 2n-side semi-regular, equilateral polygons. Further, by using these results, it has been shown that the cross-sections of regular polygons inscribed to semi-regular equilateral polygons, the vertices of equiangular semi-regular polygons, as well as the sides of the regular polygons inscribed to it, intersect in the same manner at the vertices of the equilateral semi-regular polygon. It has further been shown that the sides of the equiangular semi-regular polygon refer to each other as the sines of the angles created by the sides of the inscribed polygons and the side of the semi-regular polygon.
VL  - 8
IS  - 4
ER  -

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