To solve the problem, we need to find the angle made by a sector such that its area is one-sixth of the area of a semicircle.

1. The formula for the area of a circle is \(\pi r^2\), where \(r\) is the radius.
2. A semicircle is exactly half of a circle. Thus, the area of a semicircle is \(\frac{1}{2} \pi r^2\).
3. We know the area of the sector is one-sixth of the area of the semicircle: \(\text{Area of the Sector} = \frac{1}{6} \times \frac{1}{2} \pi r^2\).
4. The general formula for the area of a sector is \(\frac{\theta}{360^\circ} \times \pi r^2\), where \(\theta\) is the central angle of the sector in degrees.
5. Set the two expressions for the area of the sector equal to each other: \(\frac{\theta}{360^\circ} \times \pi r^2 = \frac{1}{6} \times \frac{1}{2} \pi r^2\).
6. Cancel the common terms \(\pi r^2\) from both sides: \(\frac{\theta}{360^\circ} = \frac{1}{12}\).
7. Solve for \(\theta\): \(\theta = \frac{1}{12} \times 360^\circ = 30^\circ\).

Therefore, the angle made by the sector is 30 degrees.

Hence, the correct answer is \(30^\circ\).