Perimeter of a shaded part in a circle

Question

https://cdn.discordapp.com/attachments/334723040099434498/536393147538997250/Screenshot_20190120-055431.jpg This is a question that i want to know will the solution of it be 8 pi or 8 pi + 12 , will the 2 sides of the triangle be counted in the "perimeter of the shaded part" or just the circumference

Answer 1

Notice that because the area is $36\pi$, the radius is 6. Also, $\sin \angle{ABM}=\frac {1}{2}$, $\angle{ABM}=30°$. Since $ABM$ is an isosceles triangle, $\angle{BAM}=30°$ and $\angle{AMB}=120°$. Then the larger arc is $360°-120°=240°$. Applying the circumference of a sector gives $\frac{240°}{360°}2\pi r=\frac{2}{3}*2*\pi*6=8\pi$. Adding the 2 radii from the cut-off part gives $8\pi+12$.