Two circles with same center. Average distance from point on circumference of inner circle to any point inside outer circle.

Question

We have circles A and B both having center C. Circle A contains circle B. The question is - what is the average distance between an arbitrary point on the circumference of circle B to any point inside of circle A?

Answer 1

By distance is meant a line of sight Euclidean distance.

There are regions on circle A not visible from fixed chosen point B. Beyond the horizontal chord horizon through B so to say need not be considered, they are left out blank.

Only pink areas are considered.

We have at P

$$ \bar y= \frac{\int y \;dA}{ \int dA};$$

This is the definition of area centre of gravity G that we can compute by usual methods; setting

$$ dA= \frac12 \int \rho ^2 d\theta $$

in polar coordinates etc. Reduces to calculating center of gravity G of the sector of the circle shown, getting the standard result.