# What is the average distance between two random points inside a circle?

Assume you have a circle with some radius r. What is the average distance between two random points inside the circle?

(Edit: This is different from this already answered question, because here the points are inside the circle area, not on the circle circumference.)

• Do you mean the circle interior, or the circumference? Jun 3, 2016 at 13:16
• I mean the interior Jun 3, 2016 at 13:18
• I don't think this is a duplicate of that question because that appears to be about points on the circle whereas this is about points in the circle.
– Ian
Jun 3, 2016 at 13:50
• For this problem it really matters how you are choosing the points. If you are choosing them with uniform distribution relative to a cartesian plane, or if you are choosing them with uniform radius and uniform angle. Those are not the same thing. Jul 15, 2020 at 7:47
• Does this answer your question? Average distance between two points in a circular disk Jul 15, 2020 at 7:58

Sketch. Let $S$ be the distance between the points, and let $X$ be the distance of the first point from the center of the circle.
• $F_{S|X}(s|x)=P(S \ge s | x)$
• $F_S(s)=P(S \ge s ) = \int F_{S|X}(s|x) \, f_X(x) \, dx$
• $E(S) = \int_0^\infty (1-F_S(s)) ds$