An arc AB of a circle subtends an angle x radian at centre O of the circle. If the area of the sector AOB is equal to the square of the length of the arc AB, then x is:

We know length of an arc is = angle subtended in radians * radius of the circle 

Therefore in our case Length of the arc = x*r

Also, area of sector = $$\frac{\textrm{angle subtended in radians}}{2}$$ * $$radius^2$$

Therefore in our case area of sector = $$\frac{x}{2}$$ * $$r^2$$

Also given that  area of sector = length of an arc^2

Therefore $$x^2*r^2$$ = $$\frac{x}{2}$$ * $$r^2$$

Solving we get x= 0.5

Therefore our answer is Option "A"