SOLUTION
Circumference of a circle is given by
[tex]\begin{gathered} C=2\pi r \\ \text{Where C is the circumference of the circle } \end{gathered}[/tex]
The length of an arc is given by
[tex]\begin{gathered} l=\frac{\theta}{360^o}\times2\pi r \\ \text{That is } \\ l=\frac{\theta}{360^o}\times C \end{gathered}[/tex]
From the diagram given, the arc has
[tex]\begin{gathered} A\text{ length of } \\ l=12\pi\text{ and }\theta=225^o \end{gathered}[/tex]
From here we have that
[tex]\begin{gathered} l=\frac{\theta}{360^o}\times C \\ 12\pi=\frac{225}{360^o}\times C \\ 12\pi=\frac{225C}{360} \\ 225C=360\times12\pi \\ C=\frac{360\times12\pi}{225} \\ C=\frac{96}{5}\pi \end{gathered}[/tex]
Hence the answer is
[tex]\frac{96}{5}\text{ pi}[/tex]
