Perimeter Variance of Uniform Random Triangles

TL;DR

This paper analyzes the statistical properties of the perimeter of a uniformly random triangle within a disk, deriving the density and moments of side lengths, and revealing connections to Catalan numbers.

Contribution

It provides the first known derivation of the bivariate density for two sides of a random triangle in a disk and computes key moments of the perimeter.

Findings

Expected product of two sides: approximately 0.837*R^2

Variance of perimeter: approximately 0.649*R^2

Connections to Catalan numbers

Abstract

Let T be a random triangle in a disk D of radius R (meaning that vertices are independent and uniform in D). We determine the bivariate density for two arbitrary sides a,b of T. In particular, we compute that E(a*b)=(0.837...)*R^2, which implies that Var(perimeter)=(0.649...)*R^2. No closed-form expression for either coefficient is known. The Catalan numbers also arise here.