circular arcs and a polygon

Since all arcs are identical semicircles, the blue polygon is a regular hexagon and the area is given by $A=6 \times \dfrac{\sqrt{3}}{4} \times x^2$ where $x$ is the edge length. Substitute:

$54\sqrt{3}=6 \times \dfrac{\sqrt{3}}{4} \times x^2$

$x^2=\dfrac{54 \times 4}{6} = 36$

$x=6$

So the yellow region is the sum of the areas of three semicircles with diameter $6$ .

$\text{yellow region} = 3 \times \dfrac{1}{2} \times \pi \times 3^2 = \boxed{13.5\pi}$