A Compass-Only Approach to Measuring Angles: Connecting Geometry, Number Theory, and Complex Numbers

This paper presents a simple geometric activity that begins with an elementary question: if an unknown angle is given and only a compass is available, can its measure be determined? We show that repeated copying of the angle around a circle provides a natural algorithm for measuring every rational degree angle exactly and for approximating irrational degree angles. Although the construction is purely geometric, understanding why it works introduces ideas from different areas of mathematics. The process leads to modular arithmetic, where the return of the copied ray is governed by divisibility and the greatest common divisor. This explains why the method succeeds for rational angles and why irrational angles never return exactly to the starting position. The same construction can also be interpreted algebraically by representing each rotation as multiplication by eiθ on the unit circle. From this viewpoint, repeated compass steps generate finite cyclic groups, the return condition becomes an equation involving roots of unity, and the construction connects naturally to cyclotomic polynomials and the classical theory of regular polygons. Rather than presenting new mathematical results, this paper shows how one accessible geometric problem can bridge geometry, number theory, and complex numbers. The same construction is viewed from three complementary perspectives: as a geometric algorithm, a number-theoretic explanation based on modular arithmetic, and an algebraic interpretation using roots of unity. This unified viewpoint illustrates how concepts usually taught separately can emerge naturally from a single geometric process and may help students develop a more connected understanding of mathematics.