1. **Problem statement:** We have an isosceles trapezoid with diagonal length $d$ and the diagonal forms an angle $\alpha$ with the longer base. We want to find the area of this trapezoid. 2. **Key properties and formula:** An isosceles trapezoid has two equal legs and parallel bases. The area $A$ of a trapezoid is given by: $$A = \frac{(b_1 + b_2)}{2} \times h$$ where $b_1$ and $b_2$ are the lengths of the two bases and $h$ is the height. 3. **Using the diagonal and angle:** Let the longer base be $b_1$, the shorter base be $b_2$, and the legs be equal length $l$. The diagonal $d$ forms angle $\alpha$ with the longer base $b_1$. 4. **Express height $h$ and base difference:** Drop perpendiculars from the top base to the longer base to form right triangles. The height $h$ is the vertical component of the diagonal: $$h = d \sin \alpha$$ 5. **Horizontal projection of diagonal:** The horizontal projection of the diagonal on the longer base is: $$d \cos \alpha$$ 6. **Relate bases and diagonal projection:** The difference between the bases is twice the horizontal projection of the diagonal (because the trapezoid is isosceles and the legs are equal): $$b_1 - b_2 = 2 d \cos \alpha$$ 7. **Express $b_2$ in terms of $b_1$, $d$, and $\alpha$:** $$b_2 = b_1 - 2 d \cos \alpha$$ 8. **Area formula substitution:** Substitute $b_2$ and $h$ into the area formula: $$A = \frac{b_1 + (b_1 - 2 d \cos \alpha)}{2} \times d \sin \alpha = \frac{2 b_1 - 2 d \cos \alpha}{2} \times d \sin \alpha$$ 9. **Simplify:** $$A = (b_1 - d \cos \alpha) d \sin \alpha$$ 10. **Final formula:** The area of the trapezoid is: $$\boxed{A = d \sin \alpha \left(b_1 - d \cos \alpha\right)}$$ This formula expresses the area in terms of the diagonal length $d$, the angle $\alpha$, and the longer base $b_1$.