What is the easiest and fastest method to solve this?
Draw a perpendicular from the midpoint of BC (let’s call it M) to either AC or AB. Then you know that M and A lie on the perpendicular bisector of BC. Thus, you can find r through 8 = \frac 12 r \cdot s, where s is the side length do the equilateral triangle. Finally, \frac{\pi r^2}{2} gives you the area of the semicircle.