*Originally posted by royalchicken*

**There are several different ways to interpret this, so I'm curious to see what people come up with:
**

What is the average distance from a point in or on a sphere of radius r to its center?

Hmm, If the point is selected to give an equal chance of the point being at any position within the sphere, then:

The area of a shell of radius s within the sphere is 4.pi.s^2

and the contribution of this shell of pts to the total sphere volume is

(4.pi.s^2)/(4/3.pi.r^3) = 3.s^2/r^3

to get the mean distance s we need to sum each possible s, weighted by the chance of getting that s

i.e we need to integrate 3.s^3/r^3 over all s between 0 and r

the answer is then 3/4(r^4/r^3)

= 3r/4