Two points are selected at random on a straight line segment of length 1. The argument is obviously reversible. Simplexes play a fundamental role in Algebraic Topology . Let's return to our problem. Choose n points at random on a segment of length 1.
Volumes change as the cube of the linear size.
It is easier to find segments to construct a many sided polygon than it is to find the sides of a triangle. Alexander Bogomolny 2355869 .