The dots are centers of circles and I’m solving for shaded areas.
$a=16\sqrt3$
$\pi4^2=\dfrac{\pi16}2$
$a+b=\pi8^2\cdot\dfrac16$
$a+b=\pi64\cdot\dfrac16$
$16\sqrt3+b=\pi64\cdot\dfrac16$
$b=\pi64\cdot\dfrac16-16\sqrt3$
$c=8\pi-\dfrac{32}3\pi+16\sqrt3$
$\boldsymbol{48\sqrt3-8\pi}$
The equilateral triangle $a$ has area $16\sqrt3$ as you calculated, and the three semicircles have total area $24\pi$. The complete circle has radius $\frac23×4\sqrt3=\frac8{\sqrt3}$ – its centre coincides with the triangle's centroid, which is $\frac23$ of the way from vertex to opposite median – so the circle's area is $\frac{64\pi}3$. The shaded area is therefore $16\sqrt3+24\pi-\frac{64\pi}3=16\sqrt3+\frac{8\pi}3$.