To find the area of the trapezoid, we need to determine its height. We can use the Pythagorean theorem to find the height by solving for the unknowns in the given right triangles.

1. Set up the equations using the Pythagorean theorem:
- For the right triangle with hypotenuse $25$:
$$y^2 + x^2 = 25^2$$
- For the right triangle with hypotenuse $30$:
$$y^2 + (25 - x)^2 = 30^2$$

2. **Solve the first equation for $y^2$:**
$$y^2 + x^2 = 625$$
$$y^2 = 625 - x^2$$

3. **Substitute $y^2$ from the first equation into the second equation:**
$$625 - x^2 + (25 - x)^2 = 900$$
$$625 - x^2 + 625 - 50x + x^2 = 900$$
$$1250 - 50x = 900$$

4. **Solve for $x$:**
$$1250 - 900 = 50x$$
$$350 = 50x$$
$$x = 7$$

5. **Substitute $x = 7$ back into the equation for $y^2$:**
$$y^2 = 625 - 7^2$$
$$y^2 = 625 - 49$$
$$y^2 = 576$$
$$y = 24$$

6. Calculate the area of the trapezoid:
The area $A$ of a trapezoid is given by:
$$A = \frac{1}{2} \times (b_1 + b_2) \times h$$
where $b_1$ and $b_2$ are the lengths of the parallel sides, and $h$ is the height.
$$A = \frac{1}{2} \times (11 + 36) \times 24$$
$$A = \frac{1}{2} \times 47 \times 24$$
$$A = \frac{1}{2} \times 1128$$
$$A = 564$$

The final answer is $\boxed{564}$.