jmc

Draw a square around the star figure. The side length of this square is $4$, because the side length is the diameter of the circle. The square forms $4$-quarter circles around the star figure. This is the equivalent of one large circle with radius $2$, meaning that the total area of the quarter circles is $4\pi$. The area of the square is $16$. Thus, the area of the star figure is $16-4\pi$. The area of the circle is $4\pi$. Taking the ratio of the two areas, we find the answer is $\boxed{\frac{4-\pi }{\pi }}$.