SOA Exam P Question: $P$ is a random point on the Cartesian Coordinate Plane. Find the variance of the area of a circle formed by $P$. Caution: This problem was "passed down" to me and I think the wording was altered or lost along the way.  I will post the problem as I have it and then make suggestions on what I think it should be.  I am interested if anybody has seen this problem before and knows the correct wording or if anybody can "fill in the gaps" so that the problem makes sense.
Problem: $P$ is a random point on the Cartesian Coordinate plane.  $P$ is uniformly distributed between 0 and 3.  What is the variance of the circle formed by $P$.
Edit: What is the variance of the area of the circle formed by $P$.
Here are my thoughts: It doesn't make sense to say that $P$ is uniformly distributed between 0 and 3 since $P=(x,y)$ is an ordered pair.  So this problem would only make sense if we have that $P=(x,y)$, where $X,Y$~ Uniform(0,3), or $||P||$~Uniform(0,3), where $||P||=\sqrt{x^2+y^2}$.  In either case we get different variances.
I have tried this problem under both assumptions am I am beginning to believe that it's the latter case.  If so, then I would write $A=\pi||P||^2$ and then do a random variable transformation to get the pdf of $A$.  Then I would find the first and second moments of $A$ and thus have the variance.
What do you think?  I am interested in knowing either a correct/appropriate wording of the problem or a solution (or both).
 A: To say $P$ is uniformly distributed between 0 and 3 means it's on a line, not a circle.  If both $X$ and $Y$ were uniformly distributed on (0, 3), then you'd have a square.  Also, to ask about the variance of a circle means no sense.  Do you mean the radius or the area or what?  You can't have a variance of a circle.
Next, I googled for 10 seconds and found this:
http://www.actuarialoutpost.com/actuarial_discussion_forum/showthread.php?p=2652448
I assume this is your problem.  You can then google the first few words of that problem and get tons of results with that exact problem being asked.
UPDATE: Well, let me answer a question something like yours just for your benefit.
Question: Let $R$ be uniformly distributed on $(0, 3)$.  After $R$ is chosen at random, form a circle $C$ of radius $R$.  What is the variance of the area of $C$ throughout this process?
Answer: Let $A$ be the random variable representing the area of $C$.  Then $A = \pi R^2$.  So, we need to find
$$Var(A) = E(A^2) - E(A)^2 = E(\pi^2 R^4) - E(\pi R^2)^2 = \pi^2[ E(R^4) - E(R^2)^2].$$
Thus, the problem is now simply in terms of the uniform distribution and you can find the expected values with simple integrals.  I will let you do the rest.
