Bigger circle

Area of circle is $A=\pi R^2$

For the larger circle, this is results in $55.44=\pi R^2$

which is an equation for $R$ with a solution $R=\sqrt{\dfrac{55.44}{\pi}}$ $\approx4.2$

Diameter is twice the radius, $D=2\times4.2=\boxed{8.4}$

Let $A_{1},d_{1}$ be the area and the diameter of the smaller circle and $A_{2},d_{2}$ the area and the diameter of the bigger circle.
$\frac{A_{1}}{A_{2}}=\frac{6.16}{55.44}\Rightarrow \frac{A_{1}}{A_{2}}=\frac{1}{9}\Rightarrow A_{2}=9A_{1}\Rightarrow$
$π(\frac{d_{2}}{2})^{2}=9π(\frac{d_{1}}{2})^{2}\Rightarrow (d_{2})^{2}=9(d_{1})^{2}\Rightarrow d_{2}=3d_{1}\Rightarrow \boxed{d_{2}=8.4cm}$