A (regular, convex and some weaker condition would be sufficient) polygon is a finite union of triangles with one vertex at the origin, and which only meet at their edges. (I am being ambiguous whether I consider the triangles as closed or open, but this almost surely does not make a difference.)
In case of a regular polygon, the two sides that touch the origin are equally long, even, and in the case presented in the question this side length is one and all the triangles are congruent.
That is to say, the question reduces to finding a uniformly random point in a triangle, or maybe even a triangle with extra properties. For an isosceles triangle with height h, we can choose a random height from a linear distribution that has value zero at the origin and a known value (that is uniquely determined by linearity and the total probability being one) at the wide end, and then consider a uniform distribution at the selected height.
(Bedtime and this response is CC-zero licensed, so details can be added by anyone.)
tommi
·
2023-08-26T19:33:55Z (9 months ago)