The area of a sector is one-twelfth that of the complete circle.

We have given that area of the sector is one-twelfth of the area of the circle.

$\therefore \frac{\theta}{360} \times \pi r^{2}=\frac{1}{12} \times \pi r^{2}$

Now we will simplify the above equation as below,

$\therefore \frac{\theta}{360} \times \pi r^{2}=\frac{1}{12} \times \pi r^{2}$

$\therefore \frac{\theta}{360}=\frac{1}{12}$

Now we will multiply both sides of the equation by 360.

$\therefore \theta=\frac{1}{12} \times 360$

$\therefore \theta=30$

Therefore, angle of sector is $30^{\circ}$.