ANSWER:
[tex]c=10\sqrt[]{6}\pi[/tex]STEP-BY-STEP EXPLANATION:
To calculate the area of the entire circle we must do it by means of a proportion, since the entire circle is 360 °, therefore
[tex]\frac{25\pi}{60}=\frac{x}{360}[/tex]Solving for x:
[tex]\begin{gathered} x=\frac{360\cdot25\pi}{60} \\ x=150\pi \end{gathered}[/tex]After calculating the area we can calculate the value of the radius, knowing that:
[tex]\begin{gathered} A=\pi\cdot r^2 \\ \text{solving for r} \\ 150\pi=\pi\cdot r^2 \\ r=\sqrt[]{150} \\ r=5\sqrt[]{6} \end{gathered}[/tex]Now, the formula for the circumference is the following:
[tex]\begin{gathered} c=2\pi r \\ c=2\pi\cdot5\sqrt[]{6} \\ c=10\sqrt[]{6}\pi \end{gathered}[/tex]