1. **State the problem:** We need to find the number of sides $n$ of a regular polygon whose interior angle is $150^\circ$. 2. **Formula used:** The measure of each interior angle $I$ of a regular polygon with $n$ sides is given by: $$I = \frac{(n-2) \times 180}{n}$$ 3. **Substitute the given interior angle:** $$150 = \frac{(n-2) \times 180}{n}$$ 4. **Solve for $n$:** Multiply both sides by $n$: $$150n = 180(n-2)$$ Expand the right side: $$150n = 180n - 360$$ Bring all terms to one side: $$150n - 180n = -360$$ Simplify: $$-30n = -360$$ Divide both sides by $-30$: $$n = \frac{-360}{-30} = 12$$ 5. **Interpretation:** The polygon has $12$ sides. **Final answer:** The polygon is a dodecagon with $12$ sides.