Seems like the quickest way is to recognize that the triangle is a 30-60-90 triangle and that the hypotenuse is 2x if the shorter leg is x (look up a 30-60-90 triangle on google images)
If we find x (which is 13/2 = 6.5), then we've found one of the sides of the square.
We just square it and get 42.25.
Another way of doing it is the trigonometric ratios SOH CAH TOA. We just use SOH to set up: sin(30) = OPP/HYP and rearranging it equals OPP = sin(30) x HYP which equals 6.5.
Say you don't remember the ratios of the 30-60-90 triangle or that sin(30) = 1/2; another option would be to make a full triangle out of the right triangle, so you'll get an equilateral triangle where all angles are the same degrees (60 degrees). Then, all sides of that equilateral triangle will be 13. Then, we cut down that equilateral triangle to get the right triangle we started with. And we'll end up with 13/2 = 6.5 for the length of one side of the square (then we do 6.5 x 6.5 = 42.25).
Let me know if you're having trouble visualizing this; I'll see if I can draw something up if you need it.
Check me on my reasoning and arithmetic; I may have made a typo somewhere, but the reasoning should be sound.