Some Metric Properties of Semi-Regular Equilateral Nonagons

Nenad Stojanovic

doi:doi:10.11648/j.acm.20200903.17

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Some Metric Properties of Semi-Regular Equilateral Nonagons

Nenad Stojanovic

Published in Applied and Computational Mathematics (Volume 9, Issue 3)

Received: 14 May 2020 Accepted: 2 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 nonagon. 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). Unlike regular polygons, one characteristic element is not enough to analyze the metric properties of semi-regular polygons, and an additional one is needed. To select this additional characteristic element, note that the following regular triangles can be inscribed to a semi-regular equilateral nonagon by joining vertices: ∆A₁ A₄A₇, △ A₂ A₅ A₈, △A₃ A₆ A₉. Now have a look at triangle △A₁ A₄A₇. Let us use the mark φ=∡(a,b₁) to mark the angle between side a of the semi-regular nonagon and side b₁ of the inscribed regular triangle. In interpreting the metric properties of a semi-regular equilateral nonagon, in addition to its side, we also use the angle that such side creates with the side of one of the three regular triangles that can be inscribed to such semi-regular nonagon. We consider the way in which convexity, possibility of construction, surface area, and other properties depend on a side of the semi-regular nonagon and angle φ=∡(a,b₁).

Published in	Applied and Computational Mathematics (Volume 9, Issue 3)
DOI	10.11648/j.acm.20200903.17
Page(s)	102-107
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

Polygons, Semi-Regular Nonagon. Surface Area, Convexity Poligons

References

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[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.
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[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 of Semi-Regular Equilateral Nonagons. Applied and Computational Mathematics, 9(3), 102-107. https://doi.org/10.11648/j.acm.20200903.17

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ACS Style

Nenad Stojanovic. Some Metric Properties of Semi-Regular Equilateral Nonagons. Appl. Comput. Math. 2020, 9(3), 102-107. doi: 10.11648/j.acm.20200903.17

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AMA Style

Nenad Stojanovic. Some Metric Properties of Semi-Regular Equilateral Nonagons. Appl Comput Math. 2020;9(3):102-107. doi: 10.11648/j.acm.20200903.17

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@article{10.11648/j.acm.20200903.17,
  author = {Nenad Stojanovic},
  title = {Some Metric Properties of Semi-Regular Equilateral Nonagons},
  journal = {Applied and Computational Mathematics},
  volume = {9},
  number = {3},
  pages = {102-107},
  doi = {10.11648/j.acm.20200903.17},
  url = {https://doi.org/10.11648/j.acm.20200903.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20200903.17},
  abstract = {A simple polygon that either has equal all sides or all interior angles is called a semi-regular nonagon. 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). Unlike regular polygons, one characteristic element is not enough to analyze the metric properties of semi-regular polygons, and an additional one is needed. To select this additional characteristic element, note that the following regular triangles can be inscribed to a semi-regular equilateral nonagon by joining vertices: ∆A1 A4A7, △ A2 A5 A8, △A3 A6 A9. Now have a look at triangle △A1 A4A7. Let us use the mark φ=∡(a,b1) to mark the angle between side a of the semi-regular nonagon and side b1 of the inscribed regular triangle. In interpreting the metric properties of a semi-regular equilateral nonagon, in addition to its side, we also use the angle that such side creates with the side of one of the three regular triangles that can be inscribed to such semi-regular nonagon. We consider the way in which convexity, possibility of construction, surface area, and other properties depend on a side of the semi-regular nonagon and angle φ=∡(a,b1).},
 year = {2020}
}

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TY  - JOUR
T1  - Some Metric Properties of Semi-Regular Equilateral Nonagons
AU  - Nenad Stojanovic
Y1  - 2020/06/17
PY  - 2020
N1  - https://doi.org/10.11648/j.acm.20200903.17
DO  - 10.11648/j.acm.20200903.17
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 102
EP  - 107
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20200903.17
AB  - A simple polygon that either has equal all sides or all interior angles is called a semi-regular nonagon. 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). Unlike regular polygons, one characteristic element is not enough to analyze the metric properties of semi-regular polygons, and an additional one is needed. To select this additional characteristic element, note that the following regular triangles can be inscribed to a semi-regular equilateral nonagon by joining vertices: ∆A1 A4A7, △ A2 A5 A8, △A3 A6 A9. Now have a look at triangle △A1 A4A7. Let us use the mark φ=∡(a,b1) to mark the angle between side a of the semi-regular nonagon and side b1 of the inscribed regular triangle. In interpreting the metric properties of a semi-regular equilateral nonagon, in addition to its side, we also use the angle that such side creates with the side of one of the three regular triangles that can be inscribed to such semi-regular nonagon. We consider the way in which convexity, possibility of construction, surface area, and other properties depend on a side of the semi-regular nonagon and angle φ=∡(a,b1).
VL  - 9
IS  - 3
ER  -

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Author Information

Nenad Stojanovic

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

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APA Style

Nenad Stojanovic. (2020). Some Metric Properties of Semi-Regular Equilateral Nonagons. Applied and Computational Mathematics, 9(3), 102-107. https://doi.org/10.11648/j.acm.20200903.17

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ACS Style

Nenad Stojanovic. Some Metric Properties of Semi-Regular Equilateral Nonagons. Appl. Comput. Math. 2020, 9(3), 102-107. doi: 10.11648/j.acm.20200903.17

Copy | Download

AMA Style

Nenad Stojanovic. Some Metric Properties of Semi-Regular Equilateral Nonagons. Appl Comput Math. 2020;9(3):102-107. doi: 10.11648/j.acm.20200903.17

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@article{10.11648/j.acm.20200903.17,
  author = {Nenad Stojanovic},
  title = {Some Metric Properties of Semi-Regular Equilateral Nonagons},
  journal = {Applied and Computational Mathematics},
  volume = {9},
  number = {3},
  pages = {102-107},
  doi = {10.11648/j.acm.20200903.17},
  url = {https://doi.org/10.11648/j.acm.20200903.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20200903.17},
  abstract = {A simple polygon that either has equal all sides or all interior angles is called a semi-regular nonagon. 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). Unlike regular polygons, one characteristic element is not enough to analyze the metric properties of semi-regular polygons, and an additional one is needed. To select this additional characteristic element, note that the following regular triangles can be inscribed to a semi-regular equilateral nonagon by joining vertices: ∆A1 A4A7, △ A2 A5 A8, △A3 A6 A9. Now have a look at triangle △A1 A4A7. Let us use the mark φ=∡(a,b1) to mark the angle between side a of the semi-regular nonagon and side b1 of the inscribed regular triangle. In interpreting the metric properties of a semi-regular equilateral nonagon, in addition to its side, we also use the angle that such side creates with the side of one of the three regular triangles that can be inscribed to such semi-regular nonagon. We consider the way in which convexity, possibility of construction, surface area, and other properties depend on a side of the semi-regular nonagon and angle φ=∡(a,b1).},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Some Metric Properties of Semi-Regular Equilateral Nonagons
AU  - Nenad Stojanovic
Y1  - 2020/06/17
PY  - 2020
N1  - https://doi.org/10.11648/j.acm.20200903.17
DO  - 10.11648/j.acm.20200903.17
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 102
EP  - 107
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20200903.17
AB  - A simple polygon that either has equal all sides or all interior angles is called a semi-regular nonagon. 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). Unlike regular polygons, one characteristic element is not enough to analyze the metric properties of semi-regular polygons, and an additional one is needed. To select this additional characteristic element, note that the following regular triangles can be inscribed to a semi-regular equilateral nonagon by joining vertices: ∆A1 A4A7, △ A2 A5 A8, △A3 A6 A9. Now have a look at triangle △A1 A4A7. Let us use the mark φ=∡(a,b1) to mark the angle between side a of the semi-regular nonagon and side b1 of the inscribed regular triangle. In interpreting the metric properties of a semi-regular equilateral nonagon, in addition to its side, we also use the angle that such side creates with the side of one of the three regular triangles that can be inscribed to such semi-regular nonagon. We consider the way in which convexity, possibility of construction, surface area, and other properties depend on a side of the semi-regular nonagon and angle φ=∡(a,b1).
VL  - 9
IS  - 3
ER  -

Copy | Download