Comparing Areas!!

Since the concentric center $O$ of the two circles is also the centroid of the equilateral triangle, if the radius of the orange circle is $r$ , the radius of the big circle is $2r$ .

We note that the purple area $A_p$ is one-third of the area of the big circle minus the area of the orange circle. Then $A_p = \dfrac {\pi(2r)^2 - \pi r^2}3 = \pi r^2$ . Therefore $\boxed{A_o=A_p}$ .