I will end this series of posts on generating uniformly random points inside geometric shapes by considering regular polygons. Regular polygons are equilateral and equiangular. When the number of sides is equal to 4 we have a square (it is fairly easy to generate uniformly distributed random points inside a square). As the number of sides goes to infinity we have a circle (I already showed how to generate uniformly distributed random points inside a circle here).
For all the other regular polygons with \(n\) sides you could use the technique presented in this paper to generate uniformly distributed random points. The paper is by Myron Hlynka and Deborah Loach from the Department of Mathematics and Statistics at University of Windsor in Canada.