DOI: https://doi.org/10.36347/sjpms.2026.v13i03.006

Pages: 140-149

The term "Angle Quinti-section/Penta-section" refers to the process of dividing an angle into five equal sub-angles using only a straightedge and compass within the framework of Euclidean geometry. This problem is derived from the previously published article, "An Exact and Simple Solution to the ‘Trisecting an Angle’ Problem Using Straightedge and Compass," which appeared in 2025. In ancient Greek mathematics, three classical problems significantly influenced the development of geometry: Squaring the Circle, Trisecting an Angle, and Doubling the Cube. Among these, the problem of angle quinti-section (or penta-section) specifically involves constructing an angle that is exactly one-fifth of a given arbitrary angle using only an unmarked straightedge and a compass. This thesis concentrates solely on the quinti-section/penta-section process for arbitrary angles. I present a classical straightedge-and-compass construction that achieves exact penta-section/quinti-section while avoiding the explicit use of π. This approach employs a ruler-based geometric analysis and synthesis. Although quinti-sections/penta-sections for specific angles (such as a right angle) is relatively a feeling idea straightforward, addressing the general case has not been explored within classical constraints. The concept of angle quinti-section/penta-section was formulated in December 2025, as a continuation of my research on the Angle Trisection, which was solved precisely and simply and subsequently published in SJPMS [8]. The results of this research provide an exact construction-based solution to the challenge of penta-section/quinti-section process of an angle utilizing only a straightedge and compass in Euclidean geometry. While the solution relies on theorems and corollaries from high school geometry, it remains somewhat complex. Additionally, this study introduces a novel tool, the "Penta-section Ruler," / “Regular Semi-decagon” which facilitates the quick and precise quinti-section/penta