the question is basically saying that a triangle is inscribed in a circle circumference of the circle = 2*pi*r = 4 sqrt [(pi*sqrt3)] pi * r = 2 sqrt [(pi*sqrt3)] (pi*r)/2 = sqrt [(pi*sqrt3)] [(pi*r)/2]^2 = (pi*sqrt3) pi^2 * r^2/ 4 = pi * sqrt 3 pi * r^2/4 = sqrt 3 pi * r^2 = 4 * sqrt3 ---------I probability of sand falling outside the triangle = 3/4 this means area of the circle excluding the triangle / total area of the circle = 3/4 Therefore, area of the triangle/ total area of the circle = 1-3/4 = 1/4------II area of the circle = pi * r^2 = 4 * sqrt 3 [refer I] area of the quilateral triangle = side^2 sqrt3/4 side^2 sqrt3/4 / 4 * sqrt 3 = 1/4 ---refer II side^2 / 16 = 1/4 side^2 = 16/4 side^2 = 4 side = 2 Hence E. This was an awesome question, took me about 3 mins to solve, but nonetheless brilliant.

the above post pretty much covers all the bases, but the lack of square root symbols makes it pretty hard to read. HINT: google "square root symbol", copy one off one of the pages that appears (or even from the search results - you don't even have to click the links if the √ symbol appears in the first couple of lines of text), and then paste to your heart's delight. you'll even save yourself work this way, because instead of typing "sqrt" (4 keystrokes), you can just hit ctrl-V (1 keystroke, or 2, depending on how you look at it) to paste the symbol. i would've done the same for pi, but the pi character shows up as something very much resembling an 'n' on these forums. -- if that's the circumference, then the radius is this quantity divided by 2p. (here 'p' stands for pi) which is (4√(P√3)) / 2P = 2√(P√3)) / P ** = 2√√3 / √P *** - if you don't understand this step, i'll also show the work starting from (**). starting from (***): circle area = P(r^2) = P * 4√3/P = 4√3 so triangle area = 1/4 of this = √3 starting from (**): circle area = P(r^2) = P * 4P√3 / P^2 = 4√3 so triangle area = 1/4 of this = √3