There is only one circle that passes through any three given points. Hence by suitable scaling, we can inscribe every triangle inside a unit circle of radius $1$. We define distinct triangles as triangles which have different sides regardless of of the order. Hence a triangle with sides $(a,b,c)$ and a triangle with sides $(b,c,a)$ are not distinct.
Question 1: What is the average area of all distinct triangles that can be inscribed in a unit circle?
Question 2: What is the average perimeter of all distinct triangles that can be inscribed in a unit circle?
Motivation: This question was motivated by this related question where it was proved that the average perimeter of all right triangles inscribed in a semi-circle of unit diameter is $1+4/\pi$.