Circling An N-Sided Regular Polygon with Straightedge and Compass in Euclidean Geometry

Abstract

The topic of this article means “one can construct a circle, of which area is equal exactly to a given polygon with n sides, n∈N, 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 A Circle”, all famously proven Impossible under strict compass-and-straightedge constraints, by Pierre Wantzel (1837) using field theory and algebraic methods [4], 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 the conclusions above. Among these, the "Squaring A 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 polygon with n sides, n∈N," or “Circling A Regular Polygon” for short. The circling is similar to research idea arose after the “Circling A Square” problem was studied and solved and published in “Scholars Journal of Physics, Mathematics and Statistics” 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 polygon with n sides. 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 the “Circling A Regular Polygon” problem precisely and exactly with only a straightedge and compass, without altering any premise of the problem. This independent research demonstrates the solution