  We study the structure of the unit sphere of polygonal Radon planes from a
geometric point of view. In particular, we prove that a $ 2 $-dimensional real
polygonal Banach space $ \mathbb{X} $ cannot be a Radon plane if the number of
vertices of its unit sphere is $ 4n, $ for some $ n \in \mathbb{N}. $ We next
obtain a complete characterization of polygonal Radon planes in terms of a
tractable geometric concept introduced in this article. It follows from our
characterization that every regular polygon with $ 4n+2 $ vertices, where $ n
\in \mathbb{N}, $ is the unit sphere of a Radon plane. We further give example
of a family of Radon planes for which the unit spheres are hexagons, but not
regular ones.
