The area of circle with center O is 36 and the distance from A to B

Let's start with circle O.
Since the area of circle O is 36(pi), radius is 6.
AO and BO are both radii, so they equal 6.
AB also equals 6, so ABO is an equilateral triangle, making angle AOB 60 degrees.
That means that the part of circle O that we care about (the part bulging out to the right) is the full circumference minus 1/6th of the circumference (60 degrees is 1/6th of 360). So, we want 5/6ths of the circumference of circle O.
Circumference = 2(pi)(r), so the circumference of O is 2(pi)6. We want 5/6ths of that, so we've got 2(pi)5 = 10(pi), which is roughly 31.4.

If we look at the answer choices, we can eliminate C, D, and E. Not bad. If you can't figure out what to do next, maybe you eyeball the figure and guess between A and B.

Or if you're up for it, let's prove one of those.

We need the portion of the circumference of circle P that is sticking out to the left so that we can add that amount to the 31.4 we've already got.
The full circumference of circle P is \(2(3\sqrt{2})(pi) = 6\sqrt{2}(pi)\), which is roughly 26.6.
That means half the circumference of P is roughly 13.3.
Do we want more than half of the circumference of P or less than half? More. So we need our answer to be greater than 31.4+13.3=44.7.
There's only one answer choice that's greater than 44.7: answer choice A.