1. **Problem statement:** We have a rhombus with each side length $15$ cm and an area of $40$ cm$^2$. We need to find the angles of the rhombus. 2. **Formula used:** The area of a rhombus can be calculated using the formula $$\text{Area} = \frac{1}{2} ab \sin C,$$ where $a$ and $b$ are the lengths of two adjacent sides and $C$ is the angle between them. 3. **Given values:** Since all sides of a rhombus are equal, $a = b = 15$ cm, and the area is $40$ cm$^2$. 4. **Substitute values into the formula:** $$40 = \frac{1}{2} \times 15 \times 15 \times \sin C$$ 5. **Simplify:** $$40 = \frac{1}{2} \times 225 \times \sin C = 112.5 \sin C$$ 6. **Solve for $\sin C$:** $$\sin C = \frac{40}{112.5} = \frac{8}{22.5} \approx 0.3556$$ 7. **Find angle $C$:** $$C = \arcsin(0.3556) \approx 20.8^\circ$$ 8. **Find the other angle:** Since opposite angles in a rhombus are equal and adjacent angles are supplementary, $$180^\circ - 20.8^\circ = 159.2^\circ$$ 9. **Final answer:** The angles of the rhombus are approximately $20.8^\circ$ and $159.2^\circ$. This means the rhombus has two acute angles of about $20.8^\circ$ and two obtuse angles of about $159.2^\circ$.