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Scholars Journal of Physics, Mathematics and Statistics

Profile image of Tran D I N H SonTran D I N H Son

2025, Circling the Regular Hexagon with Straightedge and Compass in Euclidean Geometry

https://doi.org/10.36347/SJPMS.2025.V12I06.003
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Abstract

Review Article The topic of this article means "one can construct a circle (O, r), of which area is equal exactly to a given regular hexagon, using a straightedge and a compass only". No scientific theory lasts forever, but specific research and discoveries continuously build upon each other. The three classic ancient Greek mathematical challenges likely referring to are "Doubling The Circle", "Trisecting An Angle" and "Squaring The Circle", all famously proven Impossible under strict compass-and-straightedge constraints, by Pierre Wantzel (1837) using field theory and algebraic methods, then also by Ferdinand von Lindemann (1882) after proving π is transcendental. These original Greek challenges remain impossible under classical rules since their proofs rely on deep algebraic/transcendental properties settled in the 19th century. Recent claims may involve reinterpretations or unrelated advances but do overturn these conclusions above. Among these, the "Squaring The Circle" problem and related problems involving π have captivated both professional and amateur mathematicians for millennia. The title of this paper refers to the concept of "constructing a circle that has the exact area of a given regular hexagon," or "Circling The Regular Hexagon", for short. This research idea arose after the "Squaring The Circle" problem was studied and solved and published in "SJPMS" in 2024 [7]. This paper presents an exact solution to constructing a circle that is concentric with and has the same area as a given regular hexagon. The solution does not rely on the number π and adheres strictly to the constraints of Euclidean Geometry, using only a straightedge and compass. The technique of "ANALYSIS" is employed to solve this "Circling The Regular Hexagon" problem precisely and exactly with only a straightedge and compass, without altering any premise of the problem. This independent research demonstrates the solution to the challenge using only these tools. All mathematical tools and propositions in this solution are derived from Euclidean geometry. The methodology involves geometric methods to arrange the given circle and its equal-area regular hexagon into a concentric position. Building on this method of exact "Circling The Regular Hexagon," one can deduce an equivalent problem to formulate a new mathematical challenge: "Regular Heptagoning The Circle" (i.e., constructing a regular heptagon that has the same area as a given circle, using only a straightedge and compass). Keywords: Circling the regular hexagon, Circle; regular hexagoning circle; circle hexagonise; Circle area of an hexagon; constructing a circle with the same area as a hexagon; Euclidean geometry; straightedge and compass.

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References (22)

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  6. Tran Dinh Son, 31/10/2024, "Exact Squaring the Circle with Straightedge and Compass Only", Scholars Journal of Physics, Mathematics and Statistics | Volume-11 | Issue-10, DOI: https://doi.org/10.36347/sjpms.2024.v11i10.004, https://www.academia.edu/125965397/Scholars_Jo urnal_of_Physics_Mathematics_and_Statistics, https://saspublishers.com/article/20927/.
  7. Tran Dinh Son, 2024, "Circling the Square with Straightedge & Compass in Euclidean Geometry" by Tran Dinh Son, in Sch J Phys Math Stat | 54-64 DOI: 10.36347/sjpms.2024.v11i05.001, https://saspublishers.com/journal- details/sjpms/149/1453/.
  8. Tran Dinh Son, 22 May 2023, "Exact Angle Trisection with Straightedge and Compass by Secondary Geometry", IJMTT 22 May 2023, https://doi.org/10.14445/22315373/IJMTT- V69I5P502 Exact Angle Trisection with Straightedge and Compass by Secondary Geometry (ijmttjournal.org) https://www.academia.edu/103490898/Exact_Angl e_Trisection_with_Straightedge_and_Compass_by _Secondary_Geometry https://www.academia.edu/ai_review/115747657.
  9. Tran Dinh Son, 13/02/2025, "Exact Doubling the Cube with Straightedge and Compass by Euclidean Geometry", IJMTT 29 August 2023, Exact Doubling The Cube with Straightedge and Compass by Euclidean Geometry (ijmttjournal.org)
  10. https://doi.org/10.14445/22315373/IJMTT- V69I8P506 https://www.semanticscholar.org/me/library/all, Sch J Phys Math Stat | 24-32, DOI: https://doi.org/10.36347/sjpms.2025.v12i02.0 01, SJPMS 13/02/2025, https://doi.org/10.36347/sjpms.2025.v12i02.001 file:///C:/Users/buung_000/Documents/Downloads/ SJPMS_122_24-32%20(3).pdf.
  11. Tran Dinh Son, 22/08/2024, Circling a Regular Pentagon with Straightedge and Compass in Euclidean Geometry,
  12. Scholars Journal of Physics, Mathematics and Statistics, Sch J Phys Math Stat ISSN 2393-8056 (Print) | ISSN 2393-8064,
  13. DOI: https://doi.org/10.36347/sjpms.2024.v11i08.001 ,
  14. Tran Dinh Son, 16/09/2024, Pentagoning the Circle with Straightedge & Compass, Sch J Phys Math Stat ISSN 2393-8056 (Print) | ISSN 2393-8064 (Online), DOI: https://doi.org/10.36347/sjpms.2024.v11i09.001 ,
  15. Tran Dinh Son, 15/03/2025, Regular Triangling a Circle with Straightedge and Compass in Euclidean Geometry, Scholars Journal Of Physics Mathematics And Statistics (SJPMS), DOI: https://doi.org/10.36347/sjpms.2025.v12i03.003
  16. Tran Dinh Son, 16/06/2025, Circling an Equilateral Triangle with Straightedge and Compass in Euclidean Geometry, Scholars Journal of Physics, Mathematics and Statistics | Volume-12 | Issue-05, Published: June 16, 2025 | 20 22
  17. DOI: https://doi.org/10.36347/sjpms.2025.v12i05.0 01, Pages: 140-148, Archive link-https://saspublishers.com/journal- details/sjpms/166/1621/, Article link - https://saspublishers.com/article/22406/.
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  22. Steve Nadis, February 8, 2022, An Ancient Geometry Problem Falls to New Mathematical Techniques, https://www.quantamagazine.org/an- ancient-geometry-problem-falls-to-new- mathematical-techniques-20220208/.

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