Hone in all 3 spots

These two graphics are all we'll need to solve this problem. The picture on the left helps us determine the radius of the largest inscribed circle with the construction of a well placed right triangle. The radius of this largest circle is $\dfrac{a\sqrt{3}}{6},$ thereby making the area of this circle equal to $\dfrac{a^2}{12}\pi.$

Now, the graphic on the right shows that the self-similar image of the large fractal is $\dfrac{1}{9}$ the size of the original. However, there are 3 such self-similar images, so the contribution by the next largest circles is $\dfrac{1}{3}$ the first contribution. It turns out that this pattern will continue infinitely, where the next largest circles are $\dfrac{1}{9}$ in size, but are in 3 times the abundance of the previous circles, meaning each area contribution is $\dfrac{1}{3}$ of the previous contribution. This pattern creates a geometric series whose first term is $\dfrac{a^2}{12}\pi$ and whose common ratio is $\dfrac{1}{3}$ . Using the geometric progression sum formula, we find that the total area is equal to

$\frac{\frac{a^2}{12}\pi}{1 - \frac{1}{3}} = \frac{a^2}{8}\pi$