This paper details work done by the authors at the Hope College Research Experiences for Undergraduates Site in Summer 2005.
Abstract: We generate triangles randomly by uniformly choosing a subset of three vertices from the vertices of a regular polygon. We determine the expected area and perimeter in terms of the number of sides of the polygon. We use combinatorial methods combined with trigonometric summation formulas arising from complex analysis. We also determine the limit of these equations to compare with a classical result on triangles whose vertices are on a circle.
This article appeared in Rose-Hulman Undergraduate Mathematics Journal 7, No. 1, 2006. You can download the published version of the paper here: PDF (last update 30 November 2006).