Find a percentage of the square is shaded in black

After all the calculations, we get area of the black region as $A_{Black}=4\times r^2(\dfrac {π}{12}+\dfrac{\sqrt 3-1}{2})\approx 4r^2\times 0.6278$ , where length of each side of the square is $2r$ . Hence the required ratio is $\eta=\dfrac{A_{Black}}{4r^2}\approx 0.6278=\boxed {62.78\%}$ .