We have the following general rule to find the length of an arc:
[tex]Arc=r\cdot\frac{\alpha\cdot(2\pi)}{360}[/tex]
where r is the radius and alpha is the measure of the central angle in degrees.
In this case, we have that r = 8 and alpha equals 270 degrees, since that is the measure of the angle that completes the full circumference, then:
[tex]\begin{gathered} Arc=8\cdot(\frac{270\cdot2\pi}{360})=8\cdot(\frac{540}{360}\pi)=8(\frac{3}{2}\pi)=\frac{24}{2}\pi=12\pi \\ \Rightarrow Arc=12\pi \end{gathered}[/tex]
therefore, the length of the gray arc is 12 pi
