Geometry Puzzle: Center of Square in Circle

if you look at the possibilities you can take one approach of dividing the square into four quadrants (like a window) and from that you can see that any point located in a diagnola opposing will make a hit on the circle. So from that we know that to pick a spot in one of the four quadrants is 1 in 4 and to pick the disagonal would be another 1 in 4 chance. As the first picks location is only relevant for the second pick then it is really just working out the odd's for the sencond being diagonal.

But there's more. It would be possible for two locations on the same level dependant upon there location to also make a circle that encompased the centre fo the square. So what about those permutations, well is you divide those squares again into four and look at it then you start thinking, this could get into some recuring details and have an urge to push square pegs into round holes phsyicaly as well as mentaly :).

That all said anything with a circle has to involve PI, even if you end up eating a entire pie just to work it out. So as the square has four sides I'm going to say the answear is 4 multiplied by PI and accept I'm probably wrong but it is what we call a educated guess.

Though you can imagine after being asked that and finding the right answear to only be asked; "Now using 3d points, what are the chances of you forming a sphere that would encompass the centre of a cube", you just know it. Also would show how adaptable there solution is.

Not the answer that anyone is looking for, but shows the importance of understanding what is asked. Edit: and learning to specify exactly what you mean when you ask a question. As the saying goes: If you do not learn to write what you mean, then you will have trouble meaning what you write.